diff --git a/Project.toml b/Project.toml
index 7e82876c5..b46d82c91 100644
--- a/Project.toml
+++ b/Project.toml
@@ -46,7 +46,7 @@ GPUArrays = "11.3.1"
 JET = "0.9, 0.10, 0.11"
 LRUCache = "1.0.2"
 LinearAlgebra = "1"
-MatrixAlgebraKit = "0.6.4"
+MatrixAlgebraKit = "0.6.5"
 Mooncake = "0.5"
 OhMyThreads = "0.8.0"
 Printf = "1"
diff --git a/ext/TensorKitMooncakeExt/linalg.jl b/ext/TensorKitMooncakeExt/linalg.jl
index 6af92f7be..a4c448015 100644
--- a/ext/TensorKitMooncakeExt/linalg.jl
+++ b/ext/TensorKitMooncakeExt/linalg.jl
@@ -86,3 +86,48 @@ function Mooncake.rrule!!(::CoDual{typeof(inv)}, A_ΔA::CoDual{<:AbstractTensorM
 
     return Ainv_ΔAinv, inv_pullback
 end
+
+# single-output projections: project_hermitian!, project_antihermitian!
+for (f!, f, adj) in (
+        (:project_hermitian!, :project_hermitian, :project_hermitian_adjoint),
+        (:project_antihermitian!, :project_antihermitian, :project_antihermitian_adjoint),
+    )
+    @eval begin
+        function Mooncake.rrule!!(f_df::CoDual{typeof($f!)}, A_dA::CoDual{<:AbstractTensorMap}, arg_darg::CoDual, alg_dalg::CoDual{<:MatrixAlgebraKit.AbstractAlgorithm})
+            A, dA = arrayify(A_dA)
+            arg, darg = A_dA === arg_darg ? (A, dA) : arrayify(arg_darg)
+
+            # don't need to copy/restore A since projections don't mutate input
+            argc = copy(arg)
+            arg = $f!(A, arg, Mooncake.primal(alg_dalg))
+
+            function $adj(::NoRData)
+                $f!(darg)
+                if dA !== darg
+                    add!(dA, darg)
+                    MatrixAlgebraKit.zero!(darg)
+                end
+                copy!(arg, argc)
+                return ntuple(Returns(NoRData()), 4)
+            end
+
+            return arg_darg, $adj
+        end
+
+        function Mooncake.rrule!!(f_df::CoDual{typeof($f)}, A_dA::CoDual{<:AbstractTensorMap}, alg_dalg::CoDual{<:MatrixAlgebraKit.AbstractAlgorithm})
+            A, dA = arrayify(A_dA)
+            output = $f(A, Mooncake.primal(alg_dalg))
+            output_doutput = Mooncake.zero_fcodual(output)
+
+            doutput = last(arrayify(output_doutput))
+            function $adj(::NoRData)
+                # TODO: need accumulating projection to avoid intermediate here
+                add!(dA, $f(doutput))
+                MatrixAlgebraKit.zero!(doutput)
+                return ntuple(Returns(NoRData()), 3)
+            end
+
+            return output_doutput, $adj
+        end
+    end
+end
diff --git a/src/factorizations/factorizations.jl b/src/factorizations/factorizations.jl
index 0c2272ac8..bac5cbcd9 100644
--- a/src/factorizations/factorizations.jl
+++ b/src/factorizations/factorizations.jl
@@ -19,8 +19,8 @@ using TensorOperations: Index2Tuple
 using MatrixAlgebraKit
 import MatrixAlgebraKit as MAK
 using MatrixAlgebraKit: AbstractAlgorithm, TruncatedAlgorithm, DiagonalAlgorithm
-using MatrixAlgebraKit: TruncationStrategy, NoTruncation, TruncationByValue,
-    TruncationByError, TruncationIntersection, TruncationByFilter, TruncationByOrder
+using MatrixAlgebraKit: TruncationStrategy, NoTruncation, TruncationByValue, TruncationByError,
+    TruncationIntersection, TruncationUnion, TruncationByFilter, TruncationByOrder
 using MatrixAlgebraKit: diagview
 
 include("utility.jl")
diff --git a/src/factorizations/truncation.jl b/src/factorizations/truncation.jl
index 007565fd0..1115f4345 100644
--- a/src/factorizations/truncation.jl
+++ b/src/factorizations/truncation.jl
@@ -265,21 +265,45 @@ function MAK.findtruncated_svd(values::SectorVector, strategy::TruncationSpace)
     return SectorDict(c => MAK.findtruncated_svd(d, blockstrategy(c)) for (c, d) in pairs(values))
 end
 
+# The implementations below assume that the `SectorDict` always contains an entry for every block sector
+# for example, if a block gets fully truncated, inds[c] = Int[].
+# This is always the case in the implementations above.
+
 function MAK.findtruncated(values::SectorVector, strategy::TruncationIntersection)
     inds = map(Base.Fix1(MAK.findtruncated, values), strategy.components)
-    return SectorDict(
-        c => mapreduce(
-                Base.Fix2(getindex, c), MatrixAlgebraKit._ind_intersect, inds
-            ) for c in intersect(map(keys, inds)...)
-    )
+    @assert _allequal(keys, inds) "missing blocks are not supported right now"
+    sectors = keys(first(inds))
+    vals = map(keys(first(inds))) do c
+        mapreduce(Base.Fix2(getindex, c), MatrixAlgebraKit._ind_intersect, inds)
+    end
+    return SectorDict{eltype(sectors), eltype(vals)}(sectors, vals)
 end
 function MAK.findtruncated_svd(values::SectorVector, strategy::TruncationIntersection)
     inds = map(Base.Fix1(MAK.findtruncated_svd, values), strategy.components)
-    return SectorDict(
-        c => mapreduce(
-                Base.Fix2(getindex, c), MatrixAlgebraKit._ind_intersect, inds
-            ) for c in intersect(map(keys, inds)...)
-    )
+    @assert _allequal(keys, inds) "missing blocks are not supported right now"
+    sectors = keys(first(inds))
+    vals = map(keys(first(inds))) do c
+        mapreduce(Base.Fix2(getindex, c), MatrixAlgebraKit._ind_intersect, inds)
+    end
+    return SectorDict{eltype(sectors), eltype(vals)}(sectors, vals)
+end
+function MAK.findtruncated(values::SectorVector, strategy::TruncationUnion)
+    inds = map(Base.Fix1(MAK.findtruncated, values), strategy.components)
+    @assert _allequal(keys, inds) "missing blocks are not supported right now"
+    sectors = keys(first(inds))
+    vals = map(keys(first(inds))) do c
+        mapreduce(Base.Fix2(getindex, c), MatrixAlgebraKit._ind_union, inds)
+    end
+    return SectorDict{eltype(sectors), eltype(vals)}(sectors, vals)
+end
+function MAK.findtruncated_svd(values::SectorVector, strategy::TruncationUnion)
+    inds = map(Base.Fix1(MAK.findtruncated_svd, values), strategy.components)
+    @assert _allequal(keys, inds) "missing blocks are not supported right now"
+    sectors = keys(first(inds))
+    vals = map(keys(first(inds))) do c
+        mapreduce(Base.Fix2(getindex, c), MatrixAlgebraKit._ind_union, inds)
+    end
+    return SectorDict{eltype(sectors), eltype(vals)}(sectors, vals)
 end
 
 # Truncation error
diff --git a/src/tensors/abstracttensor.jl b/src/tensors/abstracttensor.jl
index 2d7239460..245085d57 100644
--- a/src/tensors/abstracttensor.jl
+++ b/src/tensors/abstracttensor.jl
@@ -313,8 +313,6 @@ end
 #------------------------------------------------------------
 InnerProductStyle(t::AbstractTensorMap) = InnerProductStyle(typeof(t))
 
-blocktype(t::AbstractTensorMap) = blocktype(typeof(t))
-
 numout(t::AbstractTensorMap) = numout(typeof(t))
 numin(t::AbstractTensorMap) = numin(typeof(t))
 numind(t::AbstractTensorMap) = numind(typeof(t))
@@ -441,6 +439,7 @@ See also [`blocks`](@ref), [`blocksectors`](@ref), [`blockdim`](@ref) and [`hasb
 
 Return the type of the matrix blocks of a tensor.
 """ blocktype
+blocktype(t::AbstractTensorMap) = blocktype(typeof(t))
 function blocktype(::Type{T}) where {T <: AbstractTensorMap}
     return Core.Compiler.return_type(block, Tuple{T, sectortype(T)})
 end
diff --git a/src/tensors/tensor.jl b/src/tensors/tensor.jl
index 615c67775..598356f04 100644
--- a/src/tensors/tensor.jl
+++ b/src/tensors/tensor.jl
@@ -455,9 +455,7 @@ block(t::TensorMap, c::Sector) = blocks(t)[c]
 
 blocks(t::TensorMap) = BlockIterator(t, fusionblockstructure(t).blockstructure)
 
-function blocktype(::Type{TT}) where {TT <: TensorMap}
-    A = storagetype(TT)
-    T = eltype(A)
+function blocktype(::Type{TensorMap{T, S, N₁, N₂, A}}) where {T, S, N₁, N₂, A <: Vector{T}}
     return Base.ReshapedArray{T, 2, SubArray{T, 1, A, Tuple{UnitRange{Int}}, true}, Tuple{}}
 end
 
diff --git a/test/cuda/tensors.jl b/test/cuda/tensors.jl
index 0fad13473..7bdd90f9d 100644
--- a/test/cuda/tensors.jl
+++ b/test/cuda/tensors.jl
@@ -98,8 +98,8 @@ for V in spacelist
                 next = @constinferred Nothing iterate(bs, state)
                 b2 = @constinferred block(t, first(blocksectors(t)))
                 @test b1 == b2
-                @test_broken eltype(bs) === Pair{typeof(c), typeof(b1)}
-                @test_broken typeof(b1) === TensorKit.blocktype(t)
+                @test eltype(bs) === Pair{typeof(c), typeof(b1)}
+                @test typeof(b1) === TensorKit.blocktype(t)
                 @test typeof(c) === sectortype(t)
             end
         end
@@ -162,8 +162,8 @@ for V in spacelist
                 next = @constinferred Nothing iterate(bs, state)
                 b2 = @constinferred block(t', first(blocksectors(t')))
                 @test b1 == b2
-                @test_broken eltype(bs) === Pair{typeof(c), typeof(b1)}
-                @test_broken typeof(b1) === TensorKit.blocktype(t')
+                @test eltype(bs) === Pair{typeof(c), typeof(b1)}
+                @test typeof(b1) === TensorKit.blocktype(t')
                 @test typeof(c) === sectortype(t)
                 # linear algebra
                 @test isa(@constinferred(norm(t)), real(T))
diff --git a/test/tensors/factorizations.jl b/test/tensors/factorizations.jl
index 48dfaf6f1..de326cc13 100644
--- a/test/tensors/factorizations.jl
+++ b/test/tensors/factorizations.jl
@@ -310,6 +310,15 @@ for V in spacelist
                 @test norm(t - U5 * S5 * Vᴴ5) ≈ ϵ5 atol = eps(real(T))^(4 / 5)
                 @test minimum(diagview(S5)) >= λ
                 test_dim_isapprox(domain(S5), nvals)
+
+                trunc = truncrank(nvals) | trunctol(; atol = λ - 10eps(λ))
+                U5, S5, Vᴴ5, ϵ5 = @constinferred svd_trunc(t; trunc)
+                @test t * Vᴴ5' ≈ U5 * S5
+                @test isisometric(U5)
+                @test isisometric(Vᴴ5; side = :right)
+                @test norm(t - U5 * S5 * Vᴴ5) ≈ ϵ5 atol = eps(real(T))^(4 / 5)
+                @test minimum(diagview(S5)) >= λ
+                test_dim_isapprox(domain(S5), nvals)
             end
         end