diff --git a/Project.toml b/Project.toml
index 7e82876c5..35c8b21af 100644
--- a/Project.toml
+++ b/Project.toml
@@ -21,6 +21,7 @@ VectorInterface = "409d34a3-91d5-4945-b6ec-7529ddf182d8"
 Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
 CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+Enzyme = "7da242da-08ed-463a-9acd-ee780be4f1d9"
 FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000"
 Mooncake = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6"
 cuTENSOR = "011b41b2-24ef-40a8-b3eb-fa098493e9e1"
@@ -29,6 +30,7 @@ cuTENSOR = "011b41b2-24ef-40a8-b3eb-fa098493e9e1"
 TensorKitAdaptExt = "Adapt"
 TensorKitCUDAExt = ["CUDA", "cuTENSOR"]
 TensorKitChainRulesCoreExt = "ChainRulesCore"
+TensorKitEnzymeExt = "Enzyme"
 TensorKitFiniteDifferencesExt = "FiniteDifferences"
 TensorKitMooncakeExt = "Mooncake"
 
@@ -41,6 +43,7 @@ CUDA = "5.9"
 ChainRulesCore = "1"
 ChainRulesTestUtils = "1"
 Combinatorics = "1"
+Enzyme = "0.13.118"
 FiniteDifferences = "0.12"
 GPUArrays = "11.3.1"
 JET = "0.9, 0.10, 0.11"
@@ -73,6 +76,8 @@ CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
 ChainRulesTestUtils = "cdddcdb0-9152-4a09-a978-84456f9df70a"
 Combinatorics = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
+Enzyme = "7da242da-08ed-463a-9acd-ee780be4f1d9"
+EnzymeTestUtils = "12d8515a-0907-448a-8884-5fe00fdf1c5a"
 FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000"
 GPUArrays = "0c68f7d7-f131-5f86-a1c3-88cf8149b2d7"
 JET = "c3a54625-cd67-489e-a8e7-0a5a0ff4e31b"
@@ -86,4 +91,4 @@ Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 cuTENSOR = "011b41b2-24ef-40a8-b3eb-fa098493e9e1"
 
 [targets]
-test = ["ArgParse", "Adapt", "Aqua", "AllocCheck", "Combinatorics", "CUDA", "cuTENSOR", "GPUArrays", "LinearAlgebra", "SafeTestsets", "TensorOperations", "Test", "TestExtras", "ChainRulesCore", "ChainRulesTestUtils", "FiniteDifferences", "Zygote", "Mooncake", "JET"]
+test = ["ArgParse", "Adapt", "Aqua", "AllocCheck", "Combinatorics", "CUDA", "cuTENSOR", "GPUArrays", "LinearAlgebra", "SafeTestsets", "TensorOperations", "Test", "TestExtras", "ChainRulesCore", "ChainRulesTestUtils", "FiniteDifferences", "Zygote", "Mooncake", "Enzyme", "EnzymeTestUtils", "JET"]
diff --git a/ext/TensorKitEnzymeExt/TensorKitEnzymeExt.jl b/ext/TensorKitEnzymeExt/TensorKitEnzymeExt.jl
new file mode 100644
index 000000000..ab3061795
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/TensorKitEnzymeExt.jl
@@ -0,0 +1,21 @@
+module TensorKitEnzymeExt
+
+using Enzyme
+using TensorKit
+import TensorKit as TK
+using VectorInterface
+using TensorOperations: TensorOperations, IndexTuple, Index2Tuple, linearize
+import TensorOperations as TO
+using MatrixAlgebraKit
+using TupleTools
+using Random: AbstractRNG
+
+include("utility.jl")
+include("linalg.jl")
+include("vectorinterface.jl")
+include("tensoroperations.jl")
+include("factorizations.jl")
+include("indexmanipulations.jl")
+#include("planaroperations.jl")
+
+end
diff --git a/ext/TensorKitEnzymeExt/factorizations.jl b/ext/TensorKitEnzymeExt/factorizations.jl
new file mode 100644
index 000000000..4314f7900
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/factorizations.jl
@@ -0,0 +1,134 @@
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(MatrixAlgebraKit.copy_input)},
+        ::Type{RT},
+        cache,
+        f::Annotation,
+        A::Annotation{<:AbstractTensorMap}
+    ) where {RT}
+    copy_shadow = cache
+    if !isa(A, Const) && !isnothing(copy_shadow)
+        add!(A.dval, copy_shadow)
+    end
+    return (nothing, nothing)
+end
+
+for (f, pb) in (
+        (:eig_full, :(MatrixAlgebraKit.eig_pullback!)),
+        (:eigh_full, :(MatrixAlgebraKit.eigh_pullback!)),
+        (:lq_compact, :(MatrixAlgebraKit.lq_pullback!)),
+        (:qr_compact, :(MatrixAlgebraKit.qr_pullback!)),
+    )
+    @eval begin
+        function EnzymeRules.augmented_primal(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($f)},
+                ::Type{RT},
+                A::Annotation{<:AbstractTensorMap},
+                alg::Const,
+            ) where {RT}
+            ret = $f(A.val, alg.val)
+            dret = make_zero(ret)
+            cache = (ret, dret)
+            return EnzymeRules.AugmentedReturn(ret, dret, cache)
+        end
+        function EnzymeRules.reverse(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($f)},
+                ::Type{RT},
+                cache,
+                A::Annotation{<:AbstractTensorMap},
+                alg::Const,
+            ) where {RT}
+            ret, dret = cache
+            $pb(A.dval, A.val, ret, dret)
+            return (nothing, nothing)
+        end
+    end
+end
+
+for f in (:svd_compact, :svd_full)
+    @eval begin
+        function EnzymeRules.augmented_primal(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($f)},
+                ::Type{RT},
+                A::Annotation{<:AbstractTensorMap},
+                alg::Const,
+            ) where {RT}
+            USVᴴ = $f(A.val, alg.val)
+            dUSVᴴ = make_zero(USVᴴ)
+            cache = (USVᴴ, dUSVᴴ)
+            return EnzymeRules.AugmentedReturn(USVᴴ, dUSVᴴ, cache)
+        end
+        function EnzymeRules.reverse(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($f)},
+                ::Type{RT},
+                cache,
+                A::Annotation{<:AbstractTensorMap},
+                alg::Const,
+            ) where {RT}
+            USVᴴ, dUSVᴴ = cache
+            MatrixAlgebraKit.svd_pullback!(A.dval, A.val, USVᴴ, dUSVᴴ)
+            return (nothing, nothing)
+        end
+    end
+
+    # mutating version is not guaranteed to actually mutate
+    # so we can simply use the non-mutating version instead
+    f! = Symbol(f, :!)
+    #=@eval begin
+        function EnzymeRules.augmented_primal(
+            config::EnzymeRules.RevConfigWidth{1},
+            func::Const{typeof($f!)},
+            ::Type{RT},
+            A::Annotation{<:AbstractTensorMap},
+            USVᴴ::Annotation,
+            alg::Const,
+        ) where {RT}
+            EnzymeRules.augmented_primal(func, RT, A, alg)
+        end
+        function EnzymeRules.reverse(
+            config::EnzymeRules.RevConfigWidth{1},
+            func::Const{typeof($f!)},
+            ::Type{RT},
+            cache,
+            A::Annotation{<:AbstractTensorMap},
+            USVᴴ::Annotation,
+            alg::Const,
+        ) where {RT}
+            EnzymeRules.reverse(func, RT, A, alg)
+        end
+    end=# #hmmmm
+end
+
+# TODO
+#=
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(svd_trunc)},
+        ::Type{RT},
+        A::Annotation{<:AbstractTensorMap},
+        alg::Const,
+    ) where {RT}
+
+    USVᴴ = svd_compact(A.val, alg.val.alg)
+    USVᴴtrunc, ind = MatrixAlgebraKit.truncate(svd_trunc!, USVᴴ, alg.val.trunc)
+    ϵ = MatrixAlgebraKit.truncation_error(diagview(USVᴴ[2]), ind)
+    dUSVᴴtrunc = make_zero(USVᴴtrunc)
+    cache = (USVᴴtrunc, dUSVᴴtrunc)
+    return EnzymeRules.AugmentedReturn(USVᴴtrunc, dUSVᴴtrunc, cache) 
+end
+function EnzymeRules.reverse(
+    config::EnzymeRules.RevConfigWidth{1},
+    func::Const{typeof(svd_trunc)},
+    ::Type{RT},
+    cache,
+    A::Annotation{<:AbstractTensorMap},
+    alg::Const,
+) where {RT}
+    USVᴴ, dUSVᴴ = cache
+    MatrixAlgebraKit.svd_pullback!(A.dval, A.val, USVᴴ, dUSVᴴ)
+    return (nothing, nothing) 
+end=#
diff --git a/ext/TensorKitEnzymeExt/indexmanipulations.jl b/ext/TensorKitEnzymeExt/indexmanipulations.jl
new file mode 100644
index 000000000..02f52ef79
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/indexmanipulations.jl
@@ -0,0 +1,435 @@
+for transform in (:permute, :transpose)
+    add_transform! = Symbol(:add_, transform, :!)
+    @eval function EnzymeRules.augmented_primal(
+            config::EnzymeRules.RevConfigWidth{1},
+            func::Const{typeof(TK.$add_transform!)},
+            ::Type{RT},
+            C::Annotation{<:AbstractTensorMap},
+            A::Annotation{<:AbstractTensorMap},
+            p::Const{<:Index2Tuple},
+            α::Annotation{<:Number},
+            β::Annotation{<:Number},
+            ba::Const...
+        ) where {RT}
+        C_cache = !isa(β, Const) && EnzymeRules.overwritten(config)[2] ? copy(C.val) : nothing
+        # if we need to compute Δa, it is faster to allocate an intermediate permuted A
+        # and store that instead of repeating the permutation in the pullback each time.
+        # effectively, we replace `add_permute` by `add ∘ permute`.
+        Ap = if !isa(α, Const)
+            Ap = $transform(A.val, p.val)
+            add!(C.val, Ap.val, α.val, β.val)
+            Ap
+        else
+            TK.$add_transform!(C.val, A.val, p.val, α.val, β.val, ba.val...)
+            nothing
+        end
+        cache = (C_cache, Ap)
+        primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+        shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+        return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+    end
+    @eval function EnzymeRules.reverse(
+            config::EnzymeRules.RevConfigWidth{1},
+            func::Const{typeof(TK.$add_transform!)},
+            ::Type{RT},
+            cache,
+            C::Annotation{<:AbstractTensorMap},
+            A::Annotation{<:AbstractTensorMap},
+            p::Const{<:Index2Tuple},
+            α::Annotation{<:Number},
+            β::Annotation{<:Number},
+            ba::Const...
+        ) where {RT}
+        C_cache, Ap = cache
+        Cval = something(C_cache, C.val)
+
+        # ΔA
+        ip = invperm(linearize(p.val))
+        pΔA = _repartition(ip, A.val)
+        TC = VectorInterface.promote_scale(C.dval, α.val)
+        if scalartype(A.dval) <: Real && !(TC <: Real)
+            ΔAc = TO.tensoralloc_add(TC, C.dval, pΔA, false, Val(false))
+            TK.$add_transform!(ΔAc, C.dval, pΔA, conj(α.val), Zero(), ba.val...)
+            add!(A.dval, real(ΔAc))
+        else
+            TK.$add_transform!(A.dval, C.dval, pΔA, conj(α.val), One(), ba.val...)
+        end
+        Δαr = isnothing(Ap) ? nothing : project_scalar(α.val, inner(Ap, C.dval))
+        Δβr = pullback_dβ(C.dval, Cval, β)
+        pullback_dC!(C.dval, β.val) # this typically returns nothing
+
+        return nothing, nothing, nothing, nothing, Δαr, Δβr, map(Returns(nothing), ba)...
+    end
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(TK.add_braid!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        p::Const{<:Index2Tuple},
+        levels::Const{<:Index2Tuple},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+        ba::Const...
+    ) where {RT}
+    C_cache = !isa(β, Const) && EnzymeRules.overwritten(config)[2] ? copy(C.val) : nothing
+    # if we need to compute Δa, it is faster to allocate an intermediate braided A
+    # and store that instead of repeating the permutation in the pullback each time.
+    # effectively, we replace `add_permute` by `add ∘ permute`.
+    Ap = if !isa(α, Const)
+        Ap = braid(A.val, p.val, levels.val)
+        add!(C.val, Ap, α.val, β.val)
+        Ap
+    else
+        TK.add_braid!(C.val, A.val, p.val, levels.val, α.val, β.val, ba.val...)
+        nothing
+    end
+    cache = (C_cache, Ap)
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(TK.add_braid!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        p::Const{<:Index2Tuple},
+        levels::Const{<:Index2Tuple},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+        ba::Const...
+    ) where {RT}
+    C_cache, Ap = cache
+    Cval = something(C_cache, C.val)
+    # ΔA
+    ip = invperm(linearize(p.val))
+    pΔA = _repartition(ip, A.val)
+    ilevels = TupleTools.permute(levels.val, linearize(p.val))
+    TC = VectorInterface.promote_scale(C.dval, α.val)
+    if scalartype(A.dval) <: Real && !(TC <: Real)
+        ΔAc = TO.tensoralloc_add(TC, C.dval, pΔA, false, Val(false))
+        TK.add_braid!(ΔAc, C.dval, pΔA, ilevels, conj(α.val), Zero(), ba.val...)
+        add!(A.dval, real(ΔAc))
+    else
+        TK.add_braid!(A.dval, C.dval, pΔA, ilevels, conj(α.val), One(), ba.val...)
+    end
+    Δαr = isnothing(Ap) ? nothing : project_scalar(α.val, inner(Ap, C.dval))
+    Δβr = pullback_dβ(C.dval, C.val, β)
+    pullback_dC!(C.dval, β.val) # this typically returns nothing
+    return nothing, nothing, nothing, nothing, nothing, Δαr, Δβr, map(Returns(nothing), ba)...
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(twist!)},
+        ::Type{RT},
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    twist!(t.val, inds.val; inv = false)
+    primal = EnzymeRules.needs_primal(config) ? t.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? t.dval : nothing
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(twist!)},
+        ::Type{RT},
+        cache,
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    twist!(t.dval, inds.val; inv = true)
+    return (nothing, nothing)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(Core.kwcall)},
+        ::Type{RT},
+        kwargs::Const{@NamedTuple{inv::Bool}},
+        ::Const{typeof(twist!)},
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    inv = kwargs.val.inv
+    twist!(t.val, inds.val; inv)
+    primal = EnzymeRules.needs_primal(config) ? t.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? t.dval : nothing
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(twist!)},
+        ::Type{RT},
+        cache,
+        kwargs::Const{@NamedTuple{inv::Bool}},
+        ::Const{typeof(twist!)},
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    inv = kwargs.val.inv
+    twist!(t.dval, inds.val; inv = !inv)
+    return (nothing, nothing, nothing, nothing)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(flip)},
+        ::Type{RT},
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    t′ = flip(t.val, inds.val; inv = false)
+    dt′ = make_zero(t′)
+    primal = EnzymeRules.needs_primal(config) ? t′ : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? dt′ : nothing
+    cache = dt′
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(flip)},
+        ::Type{RT},
+        cache,
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    dt′ = cache
+    dt′′ = flip(dt′, inds.val; inv = true)
+    add!(t.dval, scalartype(t.dval) <: Real ? real(dt′′) : dt′′)
+    return (nothing, nothing)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(Core.kwcall)},
+        ::Type{RT},
+        kwargs::Const{@NamedTuple{inv::Bool}},
+        ::Const{typeof(flip)},
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    inv = kwargs.val.inv
+    t′ = flip(t.val, inds.val; inv = inv)
+    dt′ = make_zero(t′)
+    primal = EnzymeRules.needs_primal(config) ? t′ : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? dt′ : nothing
+    cache = dt′
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(Core.kwcall)},
+        ::Type{RT},
+        cache,
+        kwargs::Const{@NamedTuple{inv::Bool}},
+        ::Const{typeof(flip)},
+        t::Annotation{<:AbstractTensorMap},
+        inds::Const,
+    ) where {RT}
+    inv = kwargs.val.inv
+    dt′ = cache
+    dt′′ = flip(dt′, inds.val; inv = !inv)
+    add!(t.dval, scalartype(t.dval) <: Real ? real(dt′′) : dt′′)
+    return (nothing, nothing, nothing, nothing)
+end
+
+for insertunit in (:insertleftunit, :insertrightunit)
+    @eval begin
+        function EnzymeRules.augmented_primal(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($insertunit)},
+                ::Type{RT},
+                tsrc::Annotation{<:AbstractTensorMap},
+                ival::Const{<:Val},
+            ) where {RT}
+            tdst = $insertunit(tsrc.val, ival.val)
+            Δtdst = make_zero(tdst)
+            primal = EnzymeRules.needs_primal(config) ? tdst : nothing
+            shadow = EnzymeRules.needs_shadow(config) ? Δtdst : nothing
+            cache = (nothing, tdst, Δtdst)
+            return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+        end
+        # tdst shares data with tsrc if <:TensorMap, in this case we have to deal with correctly
+        # sharing address spaces
+        function EnzymeRules.augmented_primal(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($insertunit)},
+                ::Type{RT},
+                tsrc::Annotation{<:TensorMap},
+                ival::Const{<:Val},
+            ) where {RT}
+            tsrc_cache = copy(tsrc.val)
+            tdst = $insertunit(tsrc.val, ival.val)
+            Δtdst = $insertunit(tsrc.dval, ival.val)
+            primal = EnzymeRules.needs_primal(config) ? tdst : nothing
+            shadow = EnzymeRules.needs_shadow(config) ? Δtdst : nothing
+            return cache = (tsrc_cache, tdst, Δtdst)
+        end
+        function EnzymeRules.reverse(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof($insertunit)},
+                ::Type{RT},
+                cache,
+                tsrc::Annotation{<:AbstractTensorMap},
+                ival::Const{<:Val},
+            ) where {RT}
+            tsrc_cache, tdst, Δtdst = cache
+            # note: since data is already shared for <:TensorMap, don't have to do anything here!
+            if isnothing(tsrc_cache)
+                for (c, b) in blocks(Δtdst)
+                    add!(block(tsrc.dval, c), b)
+                end
+            end
+            return (nothing, nothing)
+        end
+        function EnzymeRules.augmented_primal(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof(Core.kwcall)},
+                ::Type{RT},
+                kwargs::Const{<:NamedTuple},
+                ::Const{typeof($insertunit)},
+                tsrc::Annotation{<:AbstractTensorMap},
+                ival::Const{<:Val},
+            ) where {RT}
+            if tsrc.val isa TensorMap && !get(kwargs.val, :copy, false)
+                tsrc_cache = copy(tsrc.val)
+                tdst = $insertunit(tsrc.val, ival.val; kwargs.val...)
+                Δtdst = $insertunit(tsrc.dval, ival.val; kwargs.val...)
+            else
+                tsrc_cache = nothing
+                tdst = $insertunit(tsrc.val, ival.val; kwargs.val...)
+                Δtdst = make_zero(tdst)
+            end
+            primal = EnzymeRules.needs_primal(config) ? tdst : nothing
+            shadow = EnzymeRules.needs_shadow(config) ? Δtdst : nothing
+            cache = (tsrc_cache, tdst, Δtdst)
+            return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+        end
+        function EnzymeRules.reverse(
+                config::EnzymeRules.RevConfigWidth{1},
+                func::Const{typeof(Core.kwcall)},
+                ::Type{RT},
+                cache,
+                kwargs::Const{<:NamedTuple},
+                ::Const{typeof($insertunit)},
+                tsrc::Annotation{<:AbstractTensorMap},
+                ival::Const{<:Val},
+            ) where {RT}
+            tsrc_cache, tdst, Δtdst = cache
+            if isnothing(tsrc_cache)
+                for (c, b) in blocks(Δtdst)
+                    add!(block(Δtsrc, c), b)
+                end
+            end
+            return (nothing, nothing, nothing, nothing)
+        end
+    end
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(removeunit)},
+        ::Type{RT},
+        tsrc::Annotation{<:AbstractTensorMap},
+        ival::Const{<:Val},
+    ) where {RT}
+    tsrc_cache = nothing
+    tdst = removeunit(tsrc.val, ival.val)
+    Δtdst = make_zero(tdst)
+    primal = EnzymeRules.needs_primal(config) ? tdst : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? Δtdst : nothing
+    cache = (tsrc_cache, tdst, Δtdst)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(removeunit)},
+        ::Type{RT},
+        tsrc::Annotation{<:TensorMap},
+        ival::Const{<:Val},
+    ) where {RT}
+    # tdst shares data with tsrc if <:TensorMap, in this case we have to deal with correctly
+    # sharing address spaces
+    tsrc_cache = copy(tsrc.val)
+    tdst = removeunit(tsrc.val, ival.val)
+    Δtdst = removeunit(tsrc.dval, ival.val)
+    primal = EnzymeRules.needs_primal(config) ? tdst : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? Δtdst : nothing
+    cache = (tsrc_cache, tdst, Δtdst)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(removeunit)},
+        ::Type{RT},
+        cache,
+        tsrc::Annotation{<:AbstractTensorMap},
+        ival::Const{<:Val},
+    ) where {RT}
+    tsrc_cache, tdst, Δtdst = cache
+    # note: since data for <: TensorMap is already shared, don't have to do anything here!
+    if isnothing(tsrc_cache)
+        for (c, b) in blocks(Δtdst)
+            add!(block(tsrc.dval, c), b)
+        end
+    end
+    return (nothing, nothing)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        ::Const{typeof(Core.kwcall)},
+        ::Type{RT},
+        kwargs::Const{<:NamedTuple},
+        ::Const{typeof(removeunit)},
+        tsrc::Annotation{<:AbstractTensorMap},
+        ival::Const{<:Val},
+    ) where {RT}
+    # tdst shares data with tsrc if <:TensorMap & copy=false, in this case we have to deal with correctly
+    # sharing address spaces
+    if tsrc.val isa TensorMap && !get(kwargs.val, :copy, false)
+        tsrc_cache = copy(tsrc.val)
+        Δtdst = removeunit(tsrc.dval, ival.val)
+    else
+        tsrc_cache = nothing
+        Δtdst = make_zero(tdst)
+    end
+    tdst = removeunit(tsrc.val, ival.val; kwargs.val...)
+    primal = EnzymeRules.needs_primal(config) ? tdst : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? Δtdst : nothing
+    cache = (tsrc_cache, tdst, Δtdst)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        ::Const{typeof(Core.kwcall)},
+        ::Type{RT},
+        cache,
+        kwargs::Const{<:NamedTuple},
+        ::Const{typeof(removeunit)},
+        tsrc::Annotation{<:AbstractTensorMap},
+        ival::Const{<:Val},
+    ) where {RT}
+    tsrc_cache, tdst, Δtdst = cache
+    # note: since data for <: TensorMap is already shared, don't have to do anything here!
+    if isnothing(tsrc_cache)
+        for (c, b) in blocks(Δtdst)
+            add!(block(tsrc.dval, c), b)
+        end
+    end
+    return (nothing, nothing, nothing, nothing)
+end
diff --git a/ext/TensorKitEnzymeExt/linalg.jl b/ext/TensorKitEnzymeExt/linalg.jl
new file mode 100644
index 000000000..c08f5ba03
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/linalg.jl
@@ -0,0 +1,140 @@
+# Shared
+# ------
+pullback_dC!(ΔC, β) = scale!(ΔC, conj(β))
+pullback_dβ(ΔC, C, β) = !isa(β, Const) ? project_scalar(β.val, inner(C, ΔC)) : nothing
+
+# Can Enzyme do this itself? Apparently not...
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(mul!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        B::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+    ) where {RT}
+    cacheC = copy(C.val)
+    AB = if !isa(α, Const)
+        AB = A.val * B.val
+        add!(C.val, AB, α.val, β.val)
+        AB
+    else
+        mul!(C.val, A.val, B.val, α.val, β.val)
+        nothing
+    end
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    cacheA = EnzymeRules.overwritten(config)[3] ? copy(A.val) : nothing
+    cacheB = EnzymeRules.overwritten(config)[4] ? copy(B.val) : nothing
+
+    cache = (cacheC, cacheA, cacheB, AB)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(mul!)},
+        ::Type{<:Const},
+        C::Const{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        B::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+    )
+    mul!(C.val, A.val, B.val, α.val, β.val)
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    return EnzymeRules.AugmentedReturn(primal, nothing, nothing)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(mul!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        B::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+    ) where {RT}
+    if RT <: Const
+        Δα = isa(α, Const) ? nothing : zero(α.val)
+        Δβ = isa(β, Const) ? nothing : zero(β.val)
+        return (nothing, nothing, nothing, Δα, Δβ)
+    end
+    cacheC, cacheA, cacheB, AB = cache
+    Cval = cacheC
+    Aval = something(cacheA, A.val)
+    Bval = something(cacheB, B.val)
+
+    !isa(A, Const) && project_mul!(A.dval, C.dval, Bval', conj(α.val))
+    !isa(B, Const) && project_mul!(B.dval, Aval', C.dval, conj(α.val))
+    Δαr = isnothing(AB) ? nothing : project_scalar(α.val, inner(AB, C.dval))
+    Δβr = pullback_dβ(C.dval, Cval, β)
+    pullback_dC!(C.dval, β.val)
+
+    return (nothing, nothing, nothing, Δαr, Δβr)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(tr)},
+        ::Type{RT},
+        A::Annotation{<:AbstractTensorMap},
+    ) where {RT}
+    ret = func.val(A.val)
+    primal = EnzymeRules.needs_primal(config) ? ret : nothing
+    cache = EnzymeRules.overwritten(config)[2] ? copy(A.val) : nothing
+    return EnzymeRules.AugmentedReturn(primal, nothing, cache)
+end
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(tr)},
+        dret::Active,
+        cache,
+        A::Annotation{<:AbstractTensorMap},
+    )
+    Aval = something(cache, A.val)
+    Δtrace = dret.val
+    if !isa(A, Const)
+        for (_, b) in blocks(A.dval)
+            TensorKit.diagview(b) .+= Δtrace
+        end
+    end
+    return (nothing,)
+end
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(tr)},
+        ::Type{<:Const},
+        cache,
+        A::Annotation{<:AbstractTensorMap},
+    )
+    return (nothing,)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(inv)},
+        ::Type{RT},
+        A::Annotation{<:AbstractTensorMap},
+    ) where {RT}
+    ret = func.val(A.val)
+    primal = EnzymeRules.needs_primal(config) ? ret : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? zero(ret) : nothing
+    cache = (copy(ret), shadow)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(inv)},
+        ::Type{RT},
+        cache,
+        A::Annotation{<:AbstractTensorMap},
+    ) where {RT}
+    Ainv, ΔAinv = cache
+    !isa(A, Const) && mul!(A.dval, Ainv' * ΔAinv, Ainv', -1, One())
+    return (nothing,)
+end
diff --git a/ext/TensorKitEnzymeExt/planaroperations.jl b/ext/TensorKitEnzymeExt/planaroperations.jl
new file mode 100644
index 000000000..effb4683c
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/planaroperations.jl
@@ -0,0 +1,80 @@
+# planartrace!
+# ------------
+# TODO: Fix planartrace pullback
+# This implementation is slightly more involved than its non-planar counterpart
+# this is because we lack a general `pAB` argument in `planarcontract`, and need
+# to keep things planar along the way.
+# In particular, we can't simply tensor product with multiple identities in one go
+# if they aren't "contiguous", e.g. p = ((1, 4, 5), ()), q = ((2, 6), (3, 7))
+
+# function Mooncake.rrule!!(
+#         ::CoDual{typeof(TensorKit.planartrace!)},
+#         C_ΔC::CoDual{<:AbstractTensorMap},
+#         A_ΔA::CoDual{<:AbstractTensorMap}, p_Δp::CoDual{<:Index2Tuple}, q_Δq::CoDual{<:Index2Tuple},
+#         α_Δα::CoDual{<:Number}, β_Δβ::CoDual{<:Number},
+#         backend_Δbackend::CoDual, allocator_Δallocator::CoDual
+#     )
+#     # prepare arguments
+#     C, ΔC = arrayify(C_ΔC)
+#     A, ΔA = arrayify(A_ΔA)
+#     p = primal(p_Δp)
+#     q = primal(q_Δq)
+#     α, β = primal.((α_Δα, β_Δβ))
+#     backend, allocator = primal.((backend_Δbackend, allocator_Δallocator))
+#
+#     # primal call
+#     C_cache = copy(C)
+#     TensorKit.planartrace!(C, A, p, q, α, β, backend, allocator)
+#
+#     function planartrace_pullback(::NoRData)
+#         copy!(C, C_cache)
+#
+#         ΔAr = planartrace_pullback_ΔA!(ΔA, ΔC, A, p, q, α, backend, allocator) # this typically returns NoRData()
+#         Δαr = planartrace_pullback_Δα(ΔC, A, p, q, α, backend, allocator)
+#         Δβr = pullback_dβ(ΔC, C, β)
+#         ΔCr = pullback_dC!(ΔC, β) # this typically returns NoRData()
+#
+#         return NoRData(),
+#             ΔCr, ΔAr, NoRData(), NoRData(),
+#             Δαr, Δβr, NoRData(), NoRData()
+#     end
+#
+#     return C_ΔC, planartrace_pullback
+# end
+
+# function planartrace_pullback_dA!(
+#         ΔA, ΔC, A, p, q, α, backend, allocator
+#     )
+#     if length(q[1]) == 0
+#         ip = invperm(linearize(p))
+#         pΔA = _repartition(ip, A)
+#         TK.add_transpose!(ΔA, ΔC, pΔA, conj(α), One(), backend, allocator)
+#         return NoRData()
+#     end
+#     # if length(q[1]) == 1
+#     #     ip = invperm((p[1]..., q[2]..., p[2]..., q[1]...))
+#     #     pdA = _repartition(ip, A)
+#     #     E = one!(TO.tensoralloc_add(scalartype(A), A, q, false))
+#     #     twist!(E, filter(x -> !isdual(space(E, x)), codomainind(E)))
+#     #     # pE = ((), trivtuple(TO.numind(q)))
+#     #     # pΔC = (trivtuple(TO.numind(p)), ())
+#     #     TensorKit.planaradd!(ΔA, ΔC ⊗ E, pdA, conj(α), One(), backend, allocator)
+#     #     return NoRData()
+#     # end
+#     error("The reverse rule for `planartrace` is not yet implemented")
+# end
+#
+# function planartrace_pullback_dα(
+#         ΔC, A, p, q, α, backend, allocator
+#     )
+#     Tdα = Mooncake.rdata_type(Mooncake.tangent_type(typeof(α)))
+#     Tdα === NoRData && return NoRData()
+#
+#     # TODO: this result might be easier to compute as:
+#     # C′ = βC + α * trace(A) ⟹ At = (C′ - βC) / α
+#     At = TO.tensoralloc_add(scalartype(A), A, p, false, Val(true), allocator)
+#     TensorKit.planartrace!(At, A, p, q, One(), Zero(), backend, allocator)
+#     Δα = project_scalar(α, inner(At, ΔC))
+#     TO.tensorfree!(At, allocator)
+#     return Δα
+# end
diff --git a/ext/TensorKitEnzymeExt/tensoroperations.jl b/ext/TensorKitEnzymeExt/tensoroperations.jl
new file mode 100644
index 000000000..661fc661e
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/tensoroperations.jl
@@ -0,0 +1,213 @@
+# tensorcontract!
+# ---------------
+# TODO: it might be beneficial to compare here if it would make sense to simply compute the
+# rrule of permute-permute-gemm-permute, rather than using the contractions directly.
+# This could possibly out save some permutations being carried out twice, at the cost of having
+# to store some more intermediate objects.
+# For example, the combination `ΔC, pΔC, false` appears in the pullback for ΔA and ΔB, so effectively
+# this permutation is done multiple times.
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(TensorKit.blas_contract!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        pA::Const{<:Index2Tuple},
+        B::Annotation{<:AbstractTensorMap},
+        pB::Const{<:Index2Tuple},
+        pAB::Const{<:Index2Tuple},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+        backend::Const,
+        allocator::Const
+    ) where {RT}
+    Ccache = isa(β, Const) ? nothing : copy(C.val)
+    A_needs_cache = EnzymeRules.overwritten(config)[3] && !(typeof(B) <: Const) && !(typeof(C) <: Const)
+    Acache = A_needs_cache ? copy(A.val) : nothing
+    B_needs_cache = EnzymeRules.overwritten(config)[5] && !(typeof(A) <: Const) && !(typeof(C) <: Const)
+    Bcache = B_needs_cache ? copy(B.val) : nothing
+    AB = if !isa(α, Const)
+        AB = TO.tensorcontract(A.val, pA.val, false, B.val, pB.val, false, pAB.val, One(), backend.val, allocator.val)
+        add!(C.val, AB, α.val, β.val)
+        AB
+    else
+        TensorKit.blas_contract!(C.val, A.val, pA.val, B.val, pB.val, pAB.val, α.val, β.val, backend.val, allocator.val)
+        nothing
+    end
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    cache = (Ccache, Acache, Bcache, AB)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(TensorKit.blas_contract!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        pA::Const{<:Index2Tuple},
+        B::Annotation{<:AbstractTensorMap},
+        pB::Const{<:Index2Tuple},
+        pAB::Const{<:Index2Tuple},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+        backend::Const,
+        allocator::Const
+    ) where {RT}
+    cacheC, cacheA, cacheB, AB = cache
+    Cval = cacheC
+    Aval = something(cacheA, A.val)
+    Bval = something(cacheB, B.val)
+
+    Δα = isnothing(AB) ? nothing : project_scalar(α.val, inner(AB, C.dval))
+    Δβ = isa(β, Const) ? nothing : pullback_dβ(C.dval, Cval, β)
+
+    if !isa(A, Const)
+        blas_contract_pullback_ΔA!(
+            A.dval, C.dval, Aval, pA.val, Bval, pB.val, pAB.val, α.val, backend.val, allocator.val
+        ) # this typically returns nothing
+    end
+    if !isa(B, Const)
+        blas_contract_pullback_ΔB!(
+            B.dval, C.dval, Aval, pA.val, Bval, pB.val, pAB.val, α.val, backend.val, allocator.val
+        ) # this typically returns nothing
+    end
+    pullback_dC!(C.dval, β.val) # this typically returns nothing
+    return nothing, nothing, nothing, nothing, nothing, nothing, Δα, Δβ, nothing, nothing
+end
+
+function blas_contract_pullback_ΔA!(
+        ΔA, ΔC, A, pA, B, pB, pAB, α, backend, allocator
+    )
+    ipAB = invperm(linearize(pAB))
+    pΔC = _repartition(ipAB, TO.numout(pA))
+    ipA = _repartition(invperm(linearize(pA)), A)
+
+    tB = twist(
+        B,
+        TupleTools.vcat(
+            filter(x -> !isdual(space(B, x)), pB[1]),
+            filter(x -> isdual(space(B, x)), pB[2])
+        ); copy = false
+    )
+
+    project_contract!(
+        ΔA,
+        ΔC, pΔC, false,
+        tB, reverse(pB), true,
+        ipA, conj(α), backend, allocator
+    )
+
+    return nothing
+end
+
+function blas_contract_pullback_ΔB!(
+        ΔB, ΔC, A, pA, B, pB, pAB, α, backend, allocator
+    )
+    ipAB = invperm(linearize(pAB))
+    pΔC = _repartition(ipAB, TO.numout(pA))
+    ipB = _repartition(invperm(linearize(pB)), B)
+
+    tA = twist(
+        A,
+        TupleTools.vcat(
+            filter(x -> isdual(space(A, x)), pA[1]),
+            filter(x -> !isdual(space(A, x)), pA[2])
+        ); copy = false
+    )
+
+    project_contract!(
+        ΔB,
+        tA, reverse(pA), true,
+        ΔC, pΔC, false,
+        ipB, conj(α), backend, allocator
+    )
+
+    return nothing
+end
+
+
+# tensortrace!
+# ------------
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(TensorKit.trace_permute!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        p::Const{<:Index2Tuple},
+        q::Const{<:Index2Tuple},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+        backend::Const,
+    ) where {RT}
+    C_cache = !isa(β, Const) ? copy(C.val) : nothing
+    A_cache = EnzymeRules.overwritten(config)[3] ? copy(A.val) : nothing
+    At = if !isa(α, Const)
+        At = TO.tensortrace(A.val, p.val, q.val, false, One(), backend.val)
+        add!(C.val, At, α.val, β.val)
+        At
+    else
+        TensorKit.trace_permute!(C.val, A.val, p.val, q.val, α.val, β.val, backend.val)
+        nothing
+    end
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    cache = (C_cache, A_cache, At)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(TensorKit.trace_permute!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        p::Const{<:Index2Tuple},
+        q::Const{<:Index2Tuple},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+        backend::Const,
+    ) where {RT}
+    C_cache, A_cache, At = cache
+    Aval = something(A_cache, A.val)
+    Cval = something(C_cache, C.val)
+    !isa(A, Const) && !isa(C, Const) && trace_permute_pullback_ΔA!(A.dval, C.dval, Aval, p.val, q.val, α.val, backend.val)
+    Δαr = if !isa(C, Const) && !isnothing(At)
+        project_scalar(α.val, inner(At, C.dval))
+    elseif !isnothing(At)
+        zero(α.val)
+    else
+        nothing
+    end
+    Δβr = if !isa(β, Const) && !isa(C, Const)
+        pullback_dβ(C.dval, Cval, β)
+    elseif !isa(β, Const)
+        zero(β.val)
+    else
+        nothing
+    end
+    !isa(C, Const) && pullback_dC!(C.dval, β.val)
+    return nothing, nothing, nothing, nothing, Δαr, Δβr, nothing
+end
+
+function trace_permute_pullback_ΔA!(
+        ΔA, ΔC, A, p, q, α, backend
+    )
+    ip = invperm((linearize(p)..., q[1]..., q[2]...))
+    pdA = _repartition(ip, A)
+    E = one!(TO.tensoralloc_add(scalartype(A), A, q, false))
+    twist!(E, filter(x -> !isdual(space(E, x)), codomainind(E)))
+    pE = ((), trivtuple(TO.numind(q)))
+    pΔC = (trivtuple(TO.numind(p)), ())
+    TO.tensorproduct!(
+        ΔA, ΔC, pΔC, false, E, pE, false, pdA, conj(α), One(), backend
+    )
+    return nothing
+end
diff --git a/ext/TensorKitEnzymeExt/utility.jl b/ext/TensorKitEnzymeExt/utility.jl
new file mode 100644
index 000000000..803432d2b
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/utility.jl
@@ -0,0 +1,67 @@
+#_needs_tangent(x) = _needs_tangent(typeof(x))
+#_needs_tangent(::Type{T}) where {T <: Number} =
+#    Mooncake.rdata_type(Mooncake.tangent_type(T)) !== NoRData
+
+# Projection
+# ----------
+"""
+    project_scalar(x::Number, dx::Number)
+
+Project a computed tangent `dx` onto the correct tangent type for `x`.
+For example, we might compute a complex `dx` but only require the real part.
+"""
+project_scalar(x::Number, dx::Number) = oftype(x, dx)
+project_scalar(x::Real, dx::Complex) = project_scalar(x, real(dx))
+
+# in-place multiplication and accumulation which might project to (real)
+# TODO: this could probably be done without allocating
+function project_mul!(C, A, B, α)
+    TC = TO.promote_contract(scalartype(A), scalartype(B), scalartype(α))
+    return if !(TC <: Real) && scalartype(C) <: Real
+        add!(C, real(mul!(zerovector(C, TC), A, B, α)))
+    else
+        mul!(C, A, B, α, One())
+    end
+end
+function project_contract!(C, A, pA, conjA, B, pB, conjB, pAB, α, backend, allocator)
+    TA = TensorKit.promote_permute(A)
+    TB = TensorKit.promote_permute(B)
+    TC = TO.promote_contract(TA, TB, scalartype(α))
+
+    return if scalartype(C) <: Real && !(TC <: Real)
+        add!(C, real(TO.tensorcontract!(zerovector(C, TC), A, pA, conjA, B, pB, conjB, pAB, α, Zero(), backend, allocator)))
+    else
+        TO.tensorcontract!(C, A, pA, conjA, B, pB, conjB, pAB, α, One(), backend, allocator)
+    end
+end
+
+# IndexTuple utility
+# ------------------
+trivtuple(N) = ntuple(identity, N)
+
+Base.@constprop :aggressive function _repartition(p::IndexTuple, N₁::Int)
+    length(p) >= N₁ ||
+        throw(ArgumentError("cannot repartition $(typeof(p)) to $N₁, $(length(p) - N₁)"))
+    return TupleTools.getindices(p, trivtuple(N₁)),
+        TupleTools.getindices(p, trivtuple(length(p) - N₁) .+ N₁)
+end
+Base.@constprop :aggressive function _repartition(p::Index2Tuple, N₁::Int)
+    return _repartition(linearize(p), N₁)
+end
+function _repartition(p::Union{IndexTuple, Index2Tuple}, ::Index2Tuple{N₁}) where {N₁}
+    return _repartition(p, N₁)
+end
+function _repartition(p::Union{IndexTuple, Index2Tuple}, t::AbstractTensorMap)
+    return _repartition(p, TensorKit.numout(t))
+end
+
+# Ignore derivatives
+# ------------------
+
+EnzymeRules.inactive(::typeof(TensorKit.fusionblockstructure), arg::Any) = nothing
+EnzymeRules.inactive(::typeof(TensorKit.select), s::HomSpace, i::Index2Tuple) = nothing
+EnzymeRules.inactive(::typeof(TensorKit.flip), s::HomSpace, i::Any) = nothing
+EnzymeRules.inactive(::typeof(TensorKit.permute), s::HomSpace, i::Index2Tuple) = nothing
+EnzymeRules.inactive(::typeof(TensorKit.braid), s::HomSpace, i::Index2Tuple, ::IndexTuple) = nothing
+EnzymeRules.inactive(::typeof(TensorKit.compose), s1::HomSpace, s2::HomSpace) = nothing
+EnzymeRules.inactive(::typeof(TensorOperations.tensorcontract), c::HomSpace, p::Index2Tuple, α::Bool, b::HomSpace, q::Index2Tuple, β::Bool, pq::Index2Tuple) = nothing
diff --git a/ext/TensorKitEnzymeExt/vectorinterface.jl b/ext/TensorKitEnzymeExt/vectorinterface.jl
new file mode 100644
index 000000000..c300e1209
--- /dev/null
+++ b/ext/TensorKitEnzymeExt/vectorinterface.jl
@@ -0,0 +1,165 @@
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(scale!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+    ) where {RT}
+    C_cache = !isa(α, Const) ? copy(C.val) : nothing
+    scale!(C.val, α.val)
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    return EnzymeRules.AugmentedReturn(primal, shadow, C_cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(scale!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+    ) where {RT}
+    Cval = something(cache, C.val)
+    Δα = if !isa(α, Const) && !isa(C, Const)
+        project_scalar(α.val, inner(Cval, C.dval))
+    elseif !isa(α, Const)
+        zero(α.val)
+    else
+        nothing
+    end
+    !isa(C, Const) && scale!(C.dval, conj(α.val))
+    return (nothing, Δα)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(scale!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+    ) where {RT}
+    A_cache = !isa(α, Const) && EnzymeRules.overwritten(config)[3] ? copy(A.val) : nothing
+    scale!(C.val, A.val, α.val)
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    cache = A_cache
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(scale!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+    ) where {RT}
+    Aval = something(cache, A.val)
+    Δα = if !isa(α, Const) && !isa(C, Const)
+        project_scalar(α.val, inner(Aval, C.dval))
+    elseif !isa(α, Const)
+        zero(α.val)
+    else
+        nothing
+    end
+    !isa(A, Const) && !isa(C, Const) && add!(A.dval, C.dval, conj(α.val))
+    !isa(C, Const) && zerovector!(C.dval)
+    return (nothing, nothing, Δα)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(add!)},
+        ::Type{RT},
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+    ) where {RT}
+    A_cache = !isa(α, Const) && EnzymeRules.overwritten(config)[3] ? copy(A.val) : nothing
+    C_cache = !isa(β, Const) ? copy(C.val) : nothing
+    add!(C.val, A.val, α.val, β.val)
+    primal = EnzymeRules.needs_primal(config) ? C.val : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? C.dval : nothing
+    cache = (A_cache, C_cache)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(add!)},
+        ::Type{RT},
+        cache,
+        C::Annotation{<:AbstractTensorMap},
+        A::Annotation{<:AbstractTensorMap},
+        α::Annotation{<:Number},
+        β::Annotation{<:Number},
+    ) where {RT}
+    A_cache, C_cache = cache
+    Aval = something(A_cache, A.val)
+    Cval = something(C_cache, C.val)
+    Δα = if !isa(α, Const) && !isa(C, Const)
+        project_scalar(α.val, inner(Aval, C.dval))
+    elseif !isa(α, Const)
+        zero(α.val)
+    else
+        nothing
+    end
+    Δβ = if !isa(β, Const) && !isa(C, Const)
+        project_scalar(β.val, inner(Cval, C.dval))
+    elseif !isa(β, Const)
+        zero(β.val)
+    else
+        nothing
+    end
+    !isa(A, Const) && !isa(C, Const) && add!(A.dval, C.dval, conj(α.val))
+    !isa(C, Const) && scale!(C.dval, conj(β.val))
+    return (nothing, nothing, Δα, Δβ)
+end
+
+function EnzymeRules.augmented_primal(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(inner)},
+        ::Type{RT},
+        A::Annotation{<:AbstractTensorMap},
+        B::Annotation{<:AbstractTensorMap},
+    ) where {RT}
+    A_cache = !isa(B, Const) && EnzymeRules.overwritten(config)[2] ? copy(A.val) : nothing
+    B_cache = !isa(A, Const) && EnzymeRules.overwritten(config)[3] ? copy(B.val) : nothing
+    ret = inner(A.val, B.val)
+    primal = EnzymeRules.needs_primal(config) ? ret : nothing
+    shadow = EnzymeRules.needs_shadow(config) ? zero(ret) : nothing
+    cache = (A_cache, B_cache)
+    return EnzymeRules.AugmentedReturn(primal, shadow, cache)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(inner)},
+        dret::Active,
+        cache,
+        A::Annotation{<:AbstractTensorMap},
+        B::Annotation{<:AbstractTensorMap},
+    )
+    A_cache, B_cache = cache
+    Aval = something(A_cache, A.val)
+    Bval = something(B_cache, B.val)
+    Δs = dret.val
+    !isa(A, Const) && add!(A.dval, Bval, conj(Δs))
+    !isa(B, Const) && add!(B.dval, Aval, Δs)
+    return (nothing, nothing)
+end
+
+function EnzymeRules.reverse(
+        config::EnzymeRules.RevConfigWidth{1},
+        func::Const{typeof(inner)},
+        ::Type{<:Const},
+        cache,
+        A::Annotation{<:AbstractTensorMap},
+        B::Annotation{<:AbstractTensorMap},
+    )
+    return (nothing, nothing)
+end
diff --git a/src/factorizations/matrixalgebrakit.jl b/src/factorizations/matrixalgebrakit.jl
index b44f20653..d742860da 100644
--- a/src/factorizations/matrixalgebrakit.jl
+++ b/src/factorizations/matrixalgebrakit.jl
@@ -43,6 +43,20 @@ for f! in (
     end
 end
 
+for f! in (:qr_compact!, :qr_full!, :lq_compact!, :lq_full!)
+    @eval function MAK.$f!(t::AbstractTensorMap, F, alg::MAK.Algorithm{:Householder})
+        foreachblock(t, F...) do _, (tblock, Fblocks...)
+            Fblocks′ = $f!(tblock, Fblocks, alg)
+            # deal with the case where the output is not in-place
+            for (b′, b) in zip(Fblocks′, Fblocks)
+                b === b′ || copy!(b, b′)
+            end
+            return nothing
+        end
+        return F
+    end
+end
+
 # Handle these separately because single output instead of tuple
 for f! in (
         :qr_null!, :lq_null!,
diff --git a/test/enzyme/factorizations.jl b/test/enzyme/factorizations.jl
new file mode 100644
index 000000000..a20eb8bb0
--- /dev/null
+++ b/test/enzyme/factorizations.jl
@@ -0,0 +1,200 @@
+using Test, TestExtras
+using TensorKit
+using TensorOperations
+using VectorInterface: Zero, One
+using MatrixAlgebraKit
+using Enzyme, EnzymeTestUtils
+using Random
+Enzyme.Compiler.VERBOSE_ERRORS[] = true
+@isdefined(TestSetup) || include("../setup.jl")
+using .TestSetup
+
+spacelist = (
+    (ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
+    (
+        Vect[FermionParity](0 => 1, 1 => 1),
+        Vect[FermionParity](0 => 1, 1 => 2)',
+        Vect[FermionParity](0 => 2, 1 => 1)',
+        Vect[FermionParity](0 => 2, 1 => 3),
+        Vect[FermionParity](0 => 2, 1 => 2),
+    ),
+    (
+        Vect[SU2Irrep](0 => 2, 1 // 2 => 1),
+        Vect[SU2Irrep](0 => 1, 1 => 1),
+        Vect[SU2Irrep](1 // 2 => 1, 1 => 1)',
+        Vect[SU2Irrep](1 // 2 => 2),
+        Vect[SU2Irrep](0 => 1, 1 // 2 => 1, 3 // 2 => 1)',
+    ), # TODO problem in svdgauge_dep
+    (
+        Vect[FibonacciAnyon](:I => 2, :τ => 1),
+        Vect[FibonacciAnyon](:I => 1, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 3),
+        Vect[FibonacciAnyon](:I => 2, :τ => 2),
+    ),
+)
+eltypes = (Float64, ComplexF64)
+
+function remove_qrgauge_dependence!(ΔQ, t, Q)
+    for (c, b) in blocks(ΔQ)
+        m, n = size(block(t, c))
+        minmn = min(m, n)
+        Qc = block(Q, c)
+        Q1 = view(Qc, 1:m, 1:minmn)
+        ΔQ2 = view(b, :, (minmn + 1):m)
+        mul!(ΔQ2, Q1, Q1' * ΔQ2)
+    end
+    return ΔQ
+end
+function remove_lqgauge_dependence!(ΔQ, t, Q)
+    for (c, b) in blocks(ΔQ)
+        m, n = size(block(t, c))
+        minmn = min(m, n)
+        Qc = block(Q, c)
+        Q1 = view(Qc, 1:minmn, 1:n)
+        ΔQ2 = view(b, (minmn + 1):n, :)
+        mul!(ΔQ2, ΔQ2 * Q1', Q1)
+    end
+    return ΔQ
+end
+function remove_eiggauge_dependence!(
+        ΔV, D, V; degeneracy_atol = MatrixAlgebraKit.default_pullback_degeneracy_atol(D)
+    )
+    gaugepart = V' * ΔV
+    for (c, b) in blocks(gaugepart)
+        Dc = diagview(block(D, c))
+        # for some reason this fails only on tests, and I cannot reproduce it in an
+        # interactive session.
+        # b[abs.(transpose(diagview(Dc)) .- diagview(Dc)) .>= degeneracy_atol] .= 0
+        for j in axes(b, 2), i in axes(b, 1)
+            abs(Dc[i] - Dc[j]) >= degeneracy_atol && (b[i, j] = 0)
+        end
+    end
+    mul!(ΔV, V / (V' * V), gaugepart, -1, 1)
+    return ΔV
+end
+function remove_eighgauge_dependence!(
+        ΔV, D, V; degeneracy_atol = MatrixAlgebraKit.default_pullback_degeneracy_atol(D)
+    )
+    gaugepart = project_antihermitian!(V' * ΔV)
+    for (c, b) in blocks(gaugepart)
+        Dc = diagview(block(D, c))
+        # for some reason this fails only on tests, and I cannot reproduce it in an
+        # interactive session.
+        # b[abs.(transpose(diagview(Dc)) .- diagview(Dc)) .>= degeneracy_atol] .= 0
+        for j in axes(b, 2), i in axes(b, 1)
+            abs(Dc[i] - Dc[j]) >= degeneracy_atol && (b[i, j] = 0)
+        end
+    end
+    mul!(ΔV, V, gaugepart, -1, 1)
+    return ΔV
+end
+function remove_svdgauge_dependence!(
+        ΔU, ΔVᴴ, U, S, Vᴴ; degeneracy_atol = MatrixAlgebraKit.default_pullback_degeneracy_atol(S)
+    )
+    UdU = U' * ΔU
+    VdV = Vᴴ * ΔVᴴ'
+    gaugepart = project_antihermitian!(UdU + VdV)
+    for (c, b) in blocks(gaugepart)
+        Sd = diagview(block(S, c))
+        # for some reason this fails only on tests, and I cannot reproduce it in an
+        # interactive session.
+        # b[abs.(transpose(diagview(Sc)) .- diagview(Sc)) .>= degeneracy_atol] .= 0
+        for j in axes(b, 2), i in axes(b, 1)
+            abs(Sd[i] - Sd[j]) >= degeneracy_atol && (b[i, j] = 0)
+        end
+    end
+    mul!(ΔU, U, gaugepart, -1, 1)
+    return ΔU, ΔVᴴ
+end
+
+@timedtestset "Enzyme - Factorizations: $(TensorKit.type_repr(sectortype(eltype(V)))) ($T)" for V in spacelist, T in eltypes
+    atol = default_tol(T)
+    rtol = default_tol(T)
+
+    @timedtestset "QR" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[1] ⊗ V[2])
+
+        EnzymeTestUtils.test_reverse(qr_compact, Duplicated, (A, Duplicated); atol, rtol)
+
+        # qr_full/qr_null requires being careful with gauges
+        QR = qr_full(A)
+        ΔQR = randn!.(similar.(QR))
+        remove_qrgauge_dependence!(ΔQR[1], A, QR[1])
+        EnzymeTestUtils.test_reverse(qr_full, Duplicated, (A, Duplicated); output_tangent = ΔQR, atol, rtol)
+        #EnzymeTestUtils.test_reverse(qr_null, Duplicated, (A, Duplicated); atol, rtol)
+
+        A = randn(T, V[1] ⊗ V[2] ← V[1])
+
+        EnzymeTestUtils.test_reverse(qr_compact, Duplicated, (A, Duplicated); atol, rtol)
+
+        # qr_full/qr_null requires being careful with gauges
+        QR = qr_full(A)
+        ΔQR = randn!.(similar.(QR))
+        remove_qrgauge_dependence!(ΔQR[1], A, QR[1])
+        EnzymeTestUtils.test_reverse(qr_full, Duplicated, (A, Duplicated); output_tangent = ΔQR, atol, rtol)
+        #EnzymeTestUtils.test_reverse(qr_null, Duplicated, (A, Duplicated); atol, rtol)
+    end
+
+    @timedtestset "LQ" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[1] ⊗ V[2])
+
+        EnzymeTestUtils.test_reverse(lq_compact, Duplicated, (A, Duplicated); atol, rtol)
+
+        # lq_full/lq_null requires being careful with gauges
+        LQ = lq_full(A)
+        ΔLQ = randn!.(similar.(LQ))
+        remove_lqgauge_dependence!(ΔLQ[2], A, LQ[2])
+        EnzymeTestUtils.test_reverse(lq_full, Duplicated, (A, Duplicated); output_tangent = ΔLQ, atol, rtol)
+        #EnzymeTestUtils.test_reverse(lq_null, Duplicated, (A, Duplicated); atol, rtol)
+
+        A = randn(T, V[1] ⊗ V[2] ← V[1])
+
+        EnzymeTestUtils.test_reverse(lq_compact, Duplicated, (A, Duplicated); atol, rtol)
+
+        # lq_full/lq_null requires being careful with gauges
+        LQ = lq_full(A)
+        ΔLQ = randn!.(similar.(LQ))
+        remove_lqgauge_dependence!(ΔLQ[2], A, LQ[2])
+        EnzymeTestUtils.test_reverse(lq_full, Duplicated, (A, Duplicated); output_tangent = ΔLQ, atol, rtol)
+        #EnzymeTestUtils.test_reverse(lq_null, Duplicated, (A, Duplicated); atol, rtol)
+    end
+
+    @timedtestset "Eigenvalue decomposition" begin
+        for t in (randn(T, V[1] ← V[1]), rand(T, V[1] ⊗ V[2] ← V[1] ⊗ V[2]))
+            DV = eig_full(t)
+            ΔDV = (DiagonalTensorMap(randn!(similar(DV[1].data)), space(DV[1])), randn!(similar(DV[2])))
+            remove_eiggauge_dependence!(ΔDV[2], DV...)
+            EnzymeTestUtils.test_reverse(eig_full, Duplicated, (t, Duplicated); output_tangent = ΔDV, atol, rtol)
+
+            th = project_hermitian(t)
+            DV = eigh_full(th)
+            ΔDV = (DiagonalTensorMap(randn!(similar(DV[1].data)), space(DV[1])), randn!(similar(DV[2])))
+            remove_eighgauge_dependence!(ΔDV[2], DV...)
+            EnzymeTestUtils.test_reverse(eigh_full ∘ project_hermitian, Duplicated, (t, Duplicated); output_tangent = ΔDV, atol, rtol)
+        end
+    end
+
+    @timedtestset "Singular value decomposition" begin
+        for t in (randn(T, V[1] ← V[1]), randn(T, V[1] ⊗ V[2] ← V[3] ⊗ V[4]))
+            USVᴴ = svd_compact(t)
+            ΔUSVᴴ = (TensorMap(randn!(similar(USVᴴ[1].data)), space(USVᴴ[1])), DiagonalTensorMap(randn!(similar(USVᴴ[2].data)), space(USVᴴ[2], 1)), TensorMap(randn!(similar(USVᴴ[3].data)), space(USVᴴ[3])))
+            remove_svdgauge_dependence!(ΔUSVᴴ[1], ΔUSVᴴ[3], USVᴴ...)
+            EnzymeTestUtils.test_reverse(svd_compact, Duplicated, (t, Duplicated); output_tangent = ΔUSVᴴ, atol, rtol)
+
+            USVᴴ = svd_full(t)
+            ΔUSVᴴ = (TensorMap(randn!(similar(USVᴴ[1].data)), space(USVᴴ[1])), TensorMap(randn!(similar(USVᴴ[2].data)), space(USVᴴ[2])), TensorMap(randn!(similar(USVᴴ[3].data)), space(USVᴴ[3])))
+            remove_svdgauge_dependence!(ΔUSVᴴ[1], ΔUSVᴴ[3], USVᴴ...)
+            EnzymeTestUtils.test_reverse(svd_full, Duplicated, (t, Duplicated); output_tangent = ΔUSVᴴ, atol, rtol)
+
+            #=V_trunc = spacetype(t)(c => min(size(b)...) ÷ 2 for (c, b) in blocks(t))
+            trunc = truncspace(V_trunc)
+            alg = MatrixAlgebraKit.select_algorithm(svd_trunc, t, nothing; trunc)
+            USVᴴtrunc = svd_trunc(t, alg)
+            ΔUSVᴴtrunc = (Enzyme.randn_tangent(Base.front(USVᴴtrunc))..., zero(last(USVᴴtrunc)))
+            remove_svdgauge_dependence!(ΔUSVᴴtrunc[1], ΔUSVᴴtrunc[3], Base.front(USVᴴtrunc)...)
+            EnzymeTestUtils.test_reverse(svd_trunc, Duplicated, (t, Duplicated), (alg, Const); output_tangent = ΔUSVᴴtrunc, atol, rtol)=#
+            # TODO
+        end
+    end
+end
diff --git a/test/enzyme/indexmanipulations.jl b/test/enzyme/indexmanipulations.jl
new file mode 100644
index 000000000..75160da16
--- /dev/null
+++ b/test/enzyme/indexmanipulations.jl
@@ -0,0 +1,153 @@
+using Test, TestExtras
+using TensorKit
+using TensorOperations
+using VectorInterface: Zero, One
+using Enzyme, EnzymeTestUtils
+using Random
+
+@isdefined(TestSetup) || include("../setup.jl")
+using .TestSetup
+
+mode = Enzyme.ReverseMode
+rng = Random.default_rng()
+
+spacelist = (
+    (ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
+    (
+        Vect[FermionParity](0 => 1, 1 => 1),
+        Vect[FermionParity](0 => 1, 1 => 2)',
+        Vect[FermionParity](0 => 2, 1 => 1)',
+        Vect[FermionParity](0 => 2, 1 => 3),
+        Vect[FermionParity](0 => 2, 1 => 2),
+    ),
+    (
+        Vect[U1Irrep](0 => 2, 1 => 1, -1 => 1),
+        Vect[U1Irrep](0 => 2, 1 => 1, -1 => 1),
+        Vect[U1Irrep](0 => 2, 1 => 2, -1 => 1)',
+        Vect[U1Irrep](0 => 1, 1 => 1, -1 => 2),
+        Vect[U1Irrep](0 => 1, 1 => 2, -1 => 1)',
+    ),
+    (
+        Vect[SU2Irrep](0 => 2, 1 // 2 => 1),
+        Vect[SU2Irrep](0 => 1, 1 => 1),
+        Vect[SU2Irrep](1 // 2 => 1, 1 => 1)',
+        Vect[SU2Irrep](1 // 2 => 2),
+        Vect[SU2Irrep](0 => 1, 1 // 2 => 1, 3 // 2 => 1)',
+    ),
+    (
+        Vect[FibonacciAnyon](:I => 2, :τ => 1),
+        Vect[FibonacciAnyon](:I => 1, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 3),
+        Vect[FibonacciAnyon](:I => 2, :τ => 2),
+    ),
+)
+eltypes = (Float64, ComplexF64)
+
+@timedtestset "Enzyme - Index Manipulations: $(TensorKit.type_repr(sectortype(eltype(V)))) ($T)" for V in spacelist, T in eltypes
+    atol = default_tol(T)
+    rtol = default_tol(T)
+    symmetricbraiding = BraidingStyle(sectortype(eltype(V))) isa SymmetricBraiding
+
+    symmetricbraiding && @timedtestset "add_permute!" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[4] ⊗ V[5])
+        α = randn(T)
+        β = randn(T)
+
+        # repeat a couple times to get some distribution of arrows
+        for _ in 1:5
+            for Tα in (Active, Const), Tβ in (Active, Const)
+                p = randindextuple(numind(A))
+                C = randn!(permute(A, p))
+                EnzymeTestUtils.test_reverse(TensorKit.add_permute!, Duplicated, (C, Duplicated), (A, Duplicated), (p, Const), (α, Tα), (β, Tβ); atol, rtol)
+                A = C
+            end
+        end
+    end
+
+    @timedtestset "add_transpose!" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[4] ⊗ V[5])
+        α = randn(T)
+        β = randn(T)
+
+        # repeat a couple times to get some distribution of arrows
+        for _ in 1:2
+            p = randcircshift(numout(A), numin(A))
+            C = randn!(transpose(A, p))
+            EnzymeTestUtils.test_reverse(TensorKit.add_transpose!, Duplicated, (C, Duplicated), (A, Duplicated), (p, Const), (One(), Const), (Zero(), Const); atol, rtol)
+            for Tα in (Const, Active), Tβ in (Const, Active)
+                EnzymeTestUtils.test_reverse(TensorKit.add_transpose!, Duplicated, (C, Duplicated), (A, Duplicated), (p, Const), (α, Tα), (β, Tβ); atol, rtol)
+                if !(T <: Real)
+                    EnzymeTestUtils.test_reverse(TensorKit.add_transpose!, Duplicated, (C, Duplicated), (real(A), Duplicated), (p, Const), (α, Tα), (β, Tβ); atol, rtol)
+                    EnzymeTestUtils.test_reverse(TensorKit.add_transpose!, Duplicated, (C, Duplicated), (A, Duplicated), (p, Const), (real(α), Tα), (β, Tβ); atol, rtol)
+                    EnzymeTestUtils.test_reverse(TensorKit.add_transpose!, Duplicated, (C, Duplicated), (real(A), Duplicated), (p, Const), (real(α), Tα), (β, Tβ); atol, rtol)
+                end
+                A = C
+            end
+        end
+    end
+
+    @timedtestset "add_braid!" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[4] ⊗ V[5])
+        α = randn(T)
+        β = randn(T)
+
+        # repeat a couple times to get some distribution of arrows
+        for _ in 1:2
+            p = randcircshift(numout(A), numin(A))
+            levels = Tuple(randperm(numind(A)))
+            C = randn!(transpose(A, p))
+            for Tα in (Active, Const), Tβ in (Active, Const)
+                EnzymeTestUtils.test_reverse(TensorKit.add_braid!, Duplicated, (C, Duplicated), (A, DuplicateD), (p, Const), (levels, Const), (α, Tα), (β, Tβ); atol, rtol)
+                if !(T <: Real)
+                    EnzymeTestUtils.test_reverse(TensorKit.add_braid!, Duplicated, (C, Duplicated), (real(A), DuplicateD), (p, Const), (levels, Const), (α, Tα), (β, Tβ); atol, rtol)
+                    EnzymeTestUtils.test_reverse(TensorKit.add_braid!, Duplicated, (C, Duplicated), (A, DuplicateD), (p, Const), (levels, Const), (real(α), Tα), (β, Tβ); atol, rtol)
+                    EnzymeTestUtils.test_reverse(TensorKit.add_braid!, Duplicated, (C, Duplicated), (A, DuplicateD), (p, Const), (levels, Const), (real(α), Tα), (real(β), Tβ); atol, rtol)
+                end
+                A = C
+            end
+        end
+    end
+
+    @timedtestset "flip_n_twist!" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[4] ⊗ V[5])
+        for TA in (Const, Duplicated)
+            if !(T <: Real && !(sectorscalartype(sectortype(A)) <: Real))
+                EnzymeTestUtils.test_reverse(twist!, TA, (A, TA), (1, Const); atol, rtol, fkwargs = (inv = false,))
+                EnzymeTestUtils.test_reverse(twist!, TA, (A, TA), ([1, 3], Const); atol, rtol, fkwargs = (inv = true,))
+                EnzymeTestUtils.test_reverse(twist!, TA, (A, TA), (1, Const); atol, rtol)
+                EnzymeTestUtils.test_reverse(twist!, TA, (A, TA), ([1, 3], Const); atol, rtol)
+            end
+
+            EnzymeTestUtils.test_reverse(flip, TA, (A, TA), (1, Const); atol, rtol, fkwargs = (inv = false,))
+            EnzymeTestUtils.test_reverse(flip, TA, (A, TA), [1, 3]; atol, rtol, fkwargs = (inv = true,))
+            EnzymeTestUtils.test_reverse(flip, TA, (A, TA), (1, Const); atol, rtol)
+            EnzymeTestUtils.test_reverse(flip, TA, (A, TA), ([1, 3], Const); atol, rtol)
+        end
+    end
+
+    @timedtestset "insert and remove units" begin
+        A = randn(T, V[1] ⊗ V[2] ← V[4] ⊗ V[5])
+
+        for TA in (Const, Duplicated)
+            for insertunit in (insertleftunit, insertrightunit)
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A, TA), (Val(1), Const); atol, rtol)
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A, TA), (Val(4), Const); atol, rtol)
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A', TA), (Val(2), Const); atol, rtol)
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A, TA), (Val(1), Const); atol, rtol, fkwargs = (copy = false,))
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A, TA), (Val(2), Const); atol, rtol, fkwargs = (copy = true,))
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A, TA), (Val(3), Const); atol, rtol, fkwargs = (copy = false, dual = true, conj = true))
+                EnzymeTestUtils.test_reverse(insertunit, TA, (A', TA), (Val(3), Const); atol, rtol, fkwargs = (copy = false, dual = true, conj = true))
+            end
+        end
+
+        for TB in (Const, Duplicated)
+            for i in 1:2
+                B = insertleftunit(A, i; dual = rand(Bool))
+                EnzymeTestUtils.test_reverse(removeunit, TB, (B, TB), (Val(i), Const); atol, rtol)
+                EnzymeTestUtils.test_reverse(removeunit, TB, (B, TB), (Val(i), Const); atol, rtol, fkwargs = (copy = false,))
+                EnzymeTestUtils.test_reverse(removeunit, TB, (B, TB), (Val(i), Const); atol, rtol, fkwargs = (copy = true,))
+            end
+        end
+    end
+end
diff --git a/test/enzyme/linalg.jl b/test/enzyme/linalg.jl
new file mode 100644
index 000000000..7781653fd
--- /dev/null
+++ b/test/enzyme/linalg.jl
@@ -0,0 +1,75 @@
+using Test, TestExtras
+using TensorKit
+using Enzyme, EnzymeTestUtils
+using Random
+
+@isdefined(TestSetup) || include("../setup.jl")
+using .TestSetup
+
+mode = Enzyme.ReverseMode
+rng = Random.default_rng()
+
+spacelist = (
+    (ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
+    (
+        Vect[FermionParity](0 => 1, 1 => 1),
+        Vect[FermionParity](0 => 1, 1 => 2)',
+        Vect[FermionParity](0 => 2, 1 => 1)',
+        Vect[FermionParity](0 => 2, 1 => 3),
+        Vect[FermionParity](0 => 2, 1 => 2),
+    ),
+    (
+        Vect[FibonacciAnyon](:I => 2, :τ => 1),
+        Vect[FibonacciAnyon](:I => 1, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 3),
+        Vect[FibonacciAnyon](:I => 2, :τ => 2),
+    ),
+    (
+        Vect[SU2Irrep](0 => 2, 1 // 2 => 1),
+        Vect[SU2Irrep](0 => 1, 1 => 1),
+        Vect[SU2Irrep](1 // 2 => 1, 1 => 1)',
+        Vect[SU2Irrep](1 // 2 => 2),
+        Vect[SU2Irrep](0 => 1, 1 // 2 => 1, 3 // 2 => 1)',
+    ),
+)
+eltypes = (Float64, ComplexF64)
+
+@timedtestset "Enzyme - LinearAlgebra: $(TensorKit.type_repr(sectortype(eltype(V)))) ($T)" for V in spacelist, T in eltypes
+    atol = default_tol(T)
+    rtol = default_tol(T)
+
+    C = randn(T, V[1] ⊗ V[2] ← V[5])
+    A = randn(T, codomain(C) ← V[3] ⊗ V[4])
+    B = randn(T, domain(A) ← domain(C))
+    α = randn(T)
+    β = randn(T)
+
+    for TC in (Const, Duplicated), TA in (Const, Duplicated), TB in (Const, Duplicated)
+        for Tα in (Active, Const), Tβ in (Active, Const)
+            EnzymeTestUtils.test_reverse(mul!, TC, (C, TC), (A, TA), (B, TB), (α, Tα), (β, Tβ); atol, rtol)
+        end
+        EnzymeTestUtils.test_reverse(mul!, TC, (C, TC), (A, TA), (B, TB); atol, rtol)
+    end
+
+    for RT in (Const, Active), TC in (Const, Duplicated)
+        EnzymeTestUtils.test_reverse(norm, RT, (C, TC), (2, Const); atol, rtol)
+        EnzymeTestUtils.test_reverse(norm, RT, (C', TC), (2, Const); atol, rtol)
+    end
+
+    D1 = randn(T, V[1] ← V[1])
+    D2 = randn(T, V[1] ⊗ V[2] ← V[1] ⊗ V[2])
+    D3 = randn(T, V[1] ⊗ V[2] ⊗ V[3] ← V[1] ⊗ V[2] ⊗ V[3])
+
+    for RT in (Const, Active), TD in (Const, Duplicated)
+        EnzymeTestUtils.test_reverse(tr, RT, (D1, TD); atol, rtol)
+        EnzymeTestUtils.test_reverse(tr, RT, (D2, TD); atol, rtol)
+        #EnzymeTestUtils.test_reverse(tr, RT, (D3, TD); atol, rtol)
+    end
+
+    for TD in (Const, Duplicated)
+        EnzymeTestUtils.test_reverse(inv, TD, (D1, TD); atol, rtol)
+        EnzymeTestUtils.test_reverse(inv, TD, (D2, TD); atol, rtol)
+        #EnzymeTestUtils.test_reverse(inv, TD, (D3, TD); atol, rtol) # TODO
+    end
+end
diff --git a/test/enzyme/planaroperations.jl b/test/enzyme/planaroperations.jl
new file mode 100644
index 000000000..fa67ea7e5
--- /dev/null
+++ b/test/enzyme/planaroperations.jl
@@ -0,0 +1,119 @@
+using Test, TestExtras
+using TensorKit
+using TensorOperations
+using VectorInterface: Zero, One
+using Mooncake
+using Random
+
+@isdefined(TestSetup) || include("../setup.jl")
+using .TestSetup
+using .TestSetup: _repartition
+
+mode = Mooncake.ReverseMode
+rng = Random.default_rng()
+
+spacelist = (
+    (ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
+    (
+        Vect[FermionParity](0 => 1, 1 => 1),
+        Vect[FermionParity](0 => 1, 1 => 2)',
+        Vect[FermionParity](0 => 2, 1 => 1)',
+        Vect[FermionParity](0 => 2, 1 => 3),
+        Vect[FermionParity](0 => 2, 1 => 2),
+    ),
+    (
+        Vect[SU2Irrep](0 => 2, 1 // 2 => 1),
+        Vect[SU2Irrep](0 => 1, 1 => 1),
+        Vect[SU2Irrep](1 // 2 => 1, 1 => 1)',
+        Vect[SU2Irrep](1 // 2 => 2),
+        Vect[SU2Irrep](0 => 1, 1 // 2 => 1, 3 // 2 => 1)',
+    ),
+    (
+        Vect[FibonacciAnyon](:I => 2, :τ => 1),
+        Vect[FibonacciAnyon](:I => 1, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 3),
+        Vect[FibonacciAnyon](:I => 2, :τ => 2),
+    ),
+)
+eltypes = (Float64, ComplexF64)
+
+@timedtestset "Mooncake - PlanarOperations: $(TensorKit.type_repr(sectortype(eltype(V)))) ($T)" for V in spacelist, T in eltypes
+    atol = default_tol(T)
+    rtol = default_tol(T)
+
+    @timedtestset "planarcontract!" begin
+        for _ in 1:5
+            d = 0
+            local V1, V2, V3, k1, k2, k3
+            # retry a couple times to make sure there are at least some nonzero elements
+            for _ in 1:10
+                k1 = rand(0:3)
+                k2 = rand(0:2)
+                k3 = rand(0:2)
+                V1 = prod(v -> rand(Bool) ? v' : v, rand(V, k1); init = one(V[1]))
+                V2 = prod(v -> rand(Bool) ? v' : v, rand(V, k2); init = one(V[1]))
+                V3 = prod(v -> rand(Bool) ? v' : v, rand(V, k3); init = one(V[1]))
+                d = min(dim(V1 ← V2), dim(V1' ← V2), dim(V2 ← V3), dim(V2' ← V3))
+                d > 1 && break
+            end
+            k′ = rand(0:(k1 + k2))
+            pA = randcircshift(k′, k1 + k2 - k′, k1)
+            ipA = _repartition(invperm(linearize(pA)), k′)
+            k′ = rand(0:(k2 + k3))
+            pB = randcircshift(k′, k2 + k3 - k′, k2)
+            ipB = _repartition(invperm(linearize(pB)), k′)
+            # TODO: primal value already is broken for this?
+            # pAB = randcircshift(k1, k3)
+            pAB = _repartition(tuple((1:(k1 + k3))...), k1)
+
+            α = randn(T)
+            β = randn(T)
+
+            A = randn(T, permute(V1 ← V2, ipA))
+            B = randn(T, permute(V2 ← V3, ipB))
+            C = randn!(
+                TensorOperations.tensoralloc_contract(
+                    T, A, pA, false, B, pB, false, pAB, Val(false)
+                )
+            )
+            Mooncake.TestUtils.test_rule(
+                rng, TensorKit.planarcontract!, C, A, pA, B, pB, pAB, One(), Zero();
+                atol, rtol, mode, is_primitive = false
+            )
+            Mooncake.TestUtils.test_rule(
+                rng, TensorKit.planarcontract!, C, A, pA, B, pB, pAB, α, β;
+                atol, rtol, mode, is_primitive = false
+            )
+        end
+    end
+
+    # TODO: currently broken
+    # @timedtestset "planartrace!" begin
+    #     for _ in 1:5
+    #         k1 = rand(0:2)
+    #         k2 = rand(0:1)
+    #         V1 = map(v -> rand(Bool) ? v' : v, rand(V, k1))
+    #         V2 = map(v -> rand(Bool) ? v' : v, rand(V, k2))
+    #         V3 = prod(x -> x ⊗ x', V2[1:k2]; init = one(V[1]))
+    #         V4 = prod(x -> x ⊗ x', V2[(k2 + 1):end]; init = one(V[1]))
+    #
+    #         k′ = rand(0:(k1 + 2k2))
+    #         (_p, _q) = randcircshift(k′, k1 + 2k2 - k′, k1)
+    #         p = _repartition(_p, rand(0:k1))
+    #         q = (tuple(_q[1:2:end]...), tuple(_q[2:2:end]...))
+    #         ip = _repartition(invperm(linearize((_p, _q))), k′)
+    #         A = randn(T, permute(prod(V1) ⊗ V3 ← V4, ip))
+    #
+    #         α = randn(T)
+    #         β = randn(T)
+    #         C = randn!(TensorOperations.tensoralloc_add(T, A, p, false, Val(false)))
+    #         Mooncake.TestUtils.test_rule(
+    #             rng, TensorKit.planartrace!,
+    #             C, A, p, q, α, β,
+    #             TensorOperations.DefaultBackend(), TensorOperations.DefaultAllocator();
+    #             atol, rtol, mode
+    #         )
+    #     end
+    # end
+end
diff --git a/test/enzyme/tensoroperations.jl b/test/enzyme/tensoroperations.jl
new file mode 100644
index 000000000..0efe4d9c8
--- /dev/null
+++ b/test/enzyme/tensoroperations.jl
@@ -0,0 +1,157 @@
+using Test, TestExtras
+using TensorKit
+using TensorOperations
+using VectorInterface: One, Zero
+using Enzyme, EnzymeTestUtils
+using Random
+
+Enzyme.Compiler.VERBOSE_ERRORS[] = true
+
+@isdefined(TestSetup) || include("../setup.jl")
+using .TestSetup
+
+mode = Enzyme.ReverseMode
+rng = Random.default_rng()
+
+spacelist = (
+    #(ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
+    (
+        Vect[FermionParity](0 => 1, 1 => 1),
+        Vect[FermionParity](0 => 1, 1 => 2)',
+        Vect[FermionParity](0 => 2, 1 => 1)',
+        Vect[FermionParity](0 => 2, 1 => 3),
+        Vect[FermionParity](0 => 2, 1 => 2),
+    ),
+    (
+        Vect[U1Irrep](0 => 2, 1 => 1, -1 => 1),
+        Vect[U1Irrep](0 => 2, 1 => 1, -1 => 1),
+        Vect[U1Irrep](0 => 2, 1 => 2, -1 => 1)',
+        Vect[U1Irrep](0 => 1, 1 => 1, -1 => 2),
+        Vect[U1Irrep](0 => 1, 1 => 2, -1 => 1)',
+    ),
+    (
+        Vect[SU2Irrep](0 => 2, 1 // 2 => 1),
+        Vect[SU2Irrep](0 => 1, 1 => 1),
+        Vect[SU2Irrep](1 // 2 => 1, 1 => 1)',
+        Vect[SU2Irrep](1 // 2 => 2),
+        Vect[SU2Irrep](0 => 1, 1 // 2 => 1, 3 // 2 => 1)',
+    ),
+    (
+        Vect[FibonacciAnyon](:I => 2, :τ => 1),
+        Vect[FibonacciAnyon](:I => 1, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 3),
+        Vect[FibonacciAnyon](:I => 2, :τ => 2),
+    ),
+)
+#eltypes = (Float64, ComplexF64)
+eltypes = (ComplexF64,)
+
+@timedtestset verbose = true "Enzyme - TensorOperations: $(TensorKit.type_repr(sectortype(eltype(V)))) ($T)" for V in spacelist, T in eltypes
+    atol = default_tol(T)
+    rtol = default_tol(T)
+    symmetricbraiding = BraidingStyle(sectortype(eltype(V))) isa SymmetricBraiding
+    @show TensorKit.type_repr(sectortype(eltype(V))), T
+    flush(stdout)
+    symmetricbraiding && @timedtestset "tensorcontract!" begin
+        for _ in 1:2
+            d = 0
+            local V1, V2, V3
+            # retry a couple times to make sure there are at least some nonzero elements
+            for _ in 1:10
+                k1 = rand(0:3)
+                k2 = rand(0:2)
+                k3 = rand(0:2)
+                V1 = prod(v -> rand(Bool) ? v' : v, rand(V, k1); init = one(V[1]))
+                V2 = prod(v -> rand(Bool) ? v' : v, rand(V, k2); init = one(V[1]))
+                V3 = prod(v -> rand(Bool) ? v' : v, rand(V, k3); init = one(V[1]))
+                d = min(dim(V1 ← V2), dim(V1' ← V2), dim(V2 ← V3), dim(V2' ← V3))
+                d > 0 && break
+            end
+            ipA = randindextuple(length(V1) + length(V2))
+            pA = _repartition(invperm(linearize(ipA)), length(V1))
+            ipB = randindextuple(length(V2) + length(V3))
+            pB = _repartition(invperm(linearize(ipB)), length(V2))
+            pAB = randindextuple(length(V1) + length(V3))
+
+            α = randn(T)
+            β = randn(T)
+            V2_conj = prod(conj, V2; init = one(V[1]))
+            A = randn(T, permute(V1 ← V2, ipA))
+            B = randn(T, permute(V2 ← V3, ipB))
+            C = randn!(
+                TensorOperations.tensoralloc_contract(
+                    T, A, pA, false, B, pB, false, pAB, Val(false)
+                )
+            )
+
+            @testset for α_ in ((One(), Const), (α, Const), (α, Active)),
+                    β_ in ((Zero(), Const), (β, Const), (β, Active))
+                EnzymeTestUtils.test_reverse(
+                    TensorKit.blas_contract!, Duplicated,
+                    (copy(C), Duplicated), (A, Duplicated), (pA, Const),
+                    (B, Duplicated), (pB, Const), (pAB, Const),
+                    α_, β_,
+                    (TensorOperations.DefaultBackend(), Const),
+                    (TensorOperations.DefaultAllocator(), Const);
+                    atol, rtol
+                )
+            end
+            if !(T <: Real)
+                EnzymeTestUtils.test_reverse(
+                    TensorKit.blas_contract!, Duplicated,
+                    (copy(C), Duplicated), (A, Duplicated), (pA, Const),
+                    (B, Duplicated), (pB, Const), (pAB, Const),
+                    (real(α), Active), (real(β), Active),
+                    (TensorOperations.DefaultBackend(), Const),
+                    (TensorOperations.DefaultAllocator(), Const);
+                    atol, rtol
+                )
+                #=EnzymeTestUtils.test_reverse(
+                    TensorKit.blas_contract!, Duplicated,
+                    (copy(C), Duplicated), (real(A), Duplicated), (pA, Const),
+                    (B, Duplicated), (pB, Const), (pAB, Const),
+                    (real(α), Active), (real(β), Active),
+                    (TensorOperations.DefaultBackend(), Const),
+                    (TensorOperations.DefaultAllocator(), Const);
+                    atol, rtol
+                )=# # TODO
+                EnzymeTestUtils.test_reverse(
+                    TensorKit.blas_contract!, Duplicated,
+                    (copy(C), Duplicated), (A, Duplicated), (pA, Const),
+                    (real(B), Duplicated), (pB, Const), (pAB, Const),
+                    (real(α), Active), (real(β), Active),
+                    (TensorOperations.DefaultBackend(), Const),
+                    (TensorOperations.DefaultAllocator(), Const);
+                    atol, rtol
+                )
+            end
+        end
+    end
+    symmetricbraiding && @timedtestset "trace_permute!" begin
+        for _ in 1:2
+            k1 = rand(0:2)
+            k2 = rand(1:2)
+            V1 = map(v -> rand(Bool) ? v' : v, rand(V, k1))
+            V2 = map(v -> rand(Bool) ? v' : v, rand(V, k2))
+
+            (_p, _q) = randindextuple(k1 + 2 * k2, k1)
+            p = _repartition(_p, rand(0:k1))
+            q = _repartition(_q, k2)
+            ip = _repartition(invperm(linearize((_p, _q))), rand(0:(k1 + 2 * k2)))
+            A = randn(T, permute(prod(V1) ⊗ prod(V2) ← prod(V2), ip))
+
+            α = randn(T)
+            β = randn(T)
+            C = randn!(TensorOperations.tensoralloc_add(T, A, p, false, Val(false)))
+            for TC in (Const, Duplicated), TA in (Const, Duplicated), Tα in (Const, Active), Tβ in (Const, Active)
+                EnzymeTestUtils.test_reverse(
+                    TensorKit.trace_permute!, TC,
+                    (copy(C), TC), (A, TA), (p, Const), (q, Const),
+                    (α, Tα), (β, Tβ), (TensorOperations.DefaultBackend(), Const);
+                    atol, rtol
+                )
+            end
+        end
+    end
+end
diff --git a/test/enzyme/vectorinterface.jl b/test/enzyme/vectorinterface.jl
new file mode 100644
index 000000000..62a0dd576
--- /dev/null
+++ b/test/enzyme/vectorinterface.jl
@@ -0,0 +1,75 @@
+using Test, TestExtras
+using TensorKit
+using TensorOperations
+using Enzyme, EnzymeTestUtils
+using Random
+
+@isdefined(TestSetup) || include("../setup.jl")
+using .TestSetup
+
+mode = Enzyme.ReverseMode
+rng = Random.default_rng()
+
+# TODO adjoints are broken!
+
+spacelist = (
+    (ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
+    (
+        Vect[FermionParity](0 => 1, 1 => 1),
+        Vect[FermionParity](0 => 1, 1 => 2)',
+        Vect[FermionParity](0 => 2, 1 => 1)',
+        Vect[FermionParity](0 => 2, 1 => 3),
+        Vect[FermionParity](0 => 2, 1 => 2),
+    ),
+    (
+        Vect[SU2Irrep](0 => 2, 1 // 2 => 1),
+        Vect[SU2Irrep](0 => 1, 1 => 1),
+        Vect[SU2Irrep](1 // 2 => 1, 1 => 1)',
+        Vect[SU2Irrep](1 // 2 => 2),
+        Vect[SU2Irrep](0 => 1, 1 // 2 => 1, 3 // 2 => 1)',
+    ),
+    (
+        Vect[FibonacciAnyon](:I => 2, :τ => 1),
+        Vect[FibonacciAnyon](:I => 1, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 2)',
+        Vect[FibonacciAnyon](:I => 2, :τ => 3),
+        Vect[FibonacciAnyon](:I => 2, :τ => 2),
+    ),
+)
+eltypes = (Float64, ComplexF64)
+
+@timedtestset "Enzyme - VectorInterface: $(TensorKit.type_repr(sectortype(eltype(V)))) ($T)" for V in spacelist, T in eltypes
+    atol = default_tol(T)
+    rtol = default_tol(T)
+
+    C = randn(T, V[1] ⊗ V[2] ← V[3] ⊗ V[4] ⊗ V[5])
+    A = randn(T, V[1] ⊗ V[2] ← V[3] ⊗ V[4] ⊗ V[5])
+    α = randn(T)
+    β = randn(T)
+
+    for TC in (Duplicated, Const), Tα in (Active, Const)
+        EnzymeTestUtils.test_reverse(scale!, TC, (C, TC), (α, Tα); atol, rtol)
+        #EnzymeTestUtils.test_reverse(scale!, TC, (C', TC), (α, Tα); atol, rtol)
+        for TA in (Duplicated, Const)
+            EnzymeTestUtils.test_reverse(scale!, TC, (C, TC), (A, TA), (α, Tα); atol, rtol)
+            #EnzymeTestUtils.test_reverse(scale!, TC, (C', TC), (A', TA), (α, Tα); atol, rtol)
+            #EnzymeTestUtils.test_reverse(scale!, TC, (copy(C'), TC), (A', TA), (α, Tα); atol, rtol)
+            #EnzymeTestUtils.test_reverse(scale!, TC, (C', TC), (copy(A'), TA), (α, Tα); atol, rtol)
+        end
+    end
+
+    for TC in (Duplicated, Const), TA in (Duplicated, Const)
+        EnzymeTestUtils.test_reverse(add!, TC, (C, TC), (A, TA); atol, rtol)
+        for Tα in (Active, Const)
+            EnzymeTestUtils.test_reverse(add!, TC, (C, TC), (A, TA), (α, Tα); atol, rtol)
+            for Tβ in (Active, Const)
+                EnzymeTestUtils.test_reverse(add!, TC, (C, TC), (A, TA), (α, Tα), (β, Tβ); atol, rtol)
+            end
+        end
+    end
+
+    for RT in (Active,), TC in (Duplicated, Const), TA in (Duplicated, Const)
+        EnzymeTestUtils.test_reverse(inner, RT, (C, TC), (A, TA); atol, rtol)
+        #EnzymeTestUtils.test_reverse(inner, RT, (C', TC), (A', TA); atol, rtol)
+    end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index ad7b4006e..9ad1f2480 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -57,11 +57,19 @@ istestfile(fn) = endswith(fn, ".jl") && !contains(fn, "setup")
 
     # somehow AD tests are unreasonably slow on Apple CI
     # and ChainRulesTestUtils doesn't like prereleases
-    if group == "chainrules" || group == "mooncake"
+    if group == "chainrules" || group == "mooncake" || group == "enzyme"
         Sys.isapple() && get(ENV, "CI", "false") == "true" && continue
         isempty(VERSION.prerelease) || continue
     end
 
+    if group == "enzyme"
+        include("enzyme/factorizations.jl")
+        include("enzyme/tensoroperations.jl")
+        include("enzyme/vectorinterface.jl")
+        include("enzyme/linalg.jl")
+        include("enzyme/indexmanipulations.jl")
+    end
+
     grouppath = joinpath(@__DIR__, group)
     @time for file in filter(istestfile, readdir(grouppath))
         @info "Running test file: $file"