diff --git a/ext/MatrixAlgebraKitChainRulesCoreExt.jl b/ext/MatrixAlgebraKitChainRulesCoreExt.jl
index 400b2a79..acfe0d83 100644
--- a/ext/MatrixAlgebraKitChainRulesCoreExt.jl
+++ b/ext/MatrixAlgebraKitChainRulesCoreExt.jl
@@ -274,4 +274,22 @@ function ChainRulesCore.rrule(::typeof(right_polar!), A, PWᴴ, alg)
     return PWᴴ, right_polar_pullback
 end
 
+function ChainRulesCore.rrule(::typeof(project_hermitian), A, alg)
+    Aₕ = project_hermitian(A, alg)
+    function project_hermitian_pullback(ΔAₕ)
+        ΔA = project_hermitian(unthunk(ΔAₕ))
+        return NoTangent(), ΔA, NoTangent()
+    end
+    return Aₕ, project_hermitian_pullback
+end
+
+function ChainRulesCore.rrule(::typeof(project_antihermitian), A, alg)
+    Aₐ = project_antihermitian(A, alg)
+    function project_antihermitian_pullback(ΔAₐ)
+        ΔA = project_antihermitian(unthunk(ΔAₐ))
+        return NoTangent(), ΔA, NoTangent()
+    end
+    return Aₐ, project_antihermitian_pullback
+end
+
 end
diff --git a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
index f32e4258..ef32d6de 100644
--- a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
+++ b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
@@ -778,4 +778,51 @@ function Mooncake.rrule!!(::CoDual{typeof(svd_trunc_no_error)}, A_dA::CoDual, al
     return USVᴴtrunc_dUSVᴴtrunc, svd_trunc_adjoint
 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
+        @is_primitive DefaultCtx Mooncake.ReverseMode Tuple{typeof($f!), Any, Any, MatrixAlgebraKit.AbstractAlgorithm}
+        function Mooncake.rrule!!(f_df::CoDual{typeof($f!)}, A_dA::CoDual, 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
+                    dA .+= darg
+                    zero!(darg)
+                end
+                copy!(arg, argc)
+                return ntuple(Returns(NoRData()), 4)
+            end
+
+            return arg_darg, $adj
+        end
+
+        @is_primitive DefaultCtx Mooncake.ReverseMode Tuple{typeof($f), Any, MatrixAlgebraKit.AbstractAlgorithm}
+        function Mooncake.rrule!!(f_df::CoDual{typeof($f)}, A_dA::CoDual, 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
+                dA .+= $f(doutput)
+                zero!(doutput)
+                return ntuple(Returns(NoRData()), 3)
+            end
+
+            return output_doutput, $adj
+        end
+    end
+end
+
 end
diff --git a/src/implementations/projections.jl b/src/implementations/projections.jl
index 857366ed..0623fea5 100644
--- a/src/implementations/projections.jl
+++ b/src/implementations/projections.jl
@@ -65,11 +65,11 @@ end
 
 function project_hermitian_native!(A::Diagonal, B::Diagonal, ::Val{anti}; kwargs...) where {anti}
     if anti
-        diagview(A) .= _imimag.(diagview(B))
+        diagview(B) .= _imimag.(diagview(A))
     else
-        diagview(A) .= real.(diagview(B))
+        diagview(B) .= real.(diagview(A))
     end
-    return A
+    return B
 end
 
 function project_hermitian_native!(A::AbstractMatrix, B::AbstractMatrix, anti::Val; blocksize = 32)
diff --git a/src/pullbacks/polar.jl b/src/pullbacks/polar.jl
index 4d498da0..a549321e 100644
--- a/src/pullbacks/polar.jl
+++ b/src/pullbacks/polar.jl
@@ -17,7 +17,7 @@ function left_polar_pullback!(ΔA::AbstractMatrix, A, WP, ΔWP; kwargs...)
     !iszerotangent(ΔW) && mul!(M, W', ΔW, 1, 1)
     !iszerotangent(ΔP) && mul!(M, ΔP, P, -1, 1)
     C = _sylvester(P, P, M' - M)
-    C .+= ΔP
+    !iszerotangent(ΔP) && (C .+= ΔP)
     ΔA = mul!(ΔA, W, C, 1, 1)
     if !iszerotangent(ΔW)
         ΔWP = ΔW / P
@@ -47,7 +47,7 @@ function right_polar_pullback!(ΔA::AbstractMatrix, A, PWᴴ, ΔPWᴴ; kwargs...
     !iszerotangent(ΔWᴴ) && mul!(M, ΔWᴴ, Wᴴ', 1, 1)
     !iszerotangent(ΔP) && mul!(M, P, ΔP, -1, 1)
     C = _sylvester(P, P, M' - M)
-    C .+= ΔP
+    !iszerotangent(ΔP) && (C .+= ΔP)
     ΔA = mul!(ΔA, C, Wᴴ, 1, 1)
     if !iszerotangent(ΔWᴴ)
         PΔWᴴ = P \ ΔWᴴ
diff --git a/test/mooncake/projections.jl b/test/mooncake/projections.jl
new file mode 100644
index 00000000..b33967e9
--- /dev/null
+++ b/test/mooncake/projections.jl
@@ -0,0 +1,21 @@
+using MatrixAlgebraKit
+using Test
+using LinearAlgebra: Diagonal
+using CUDA, AMDGPU
+
+BLASFloats = (Float32, ComplexF64) # full suite is too expensive on CI
+GenericFloats = ()
+@isdefined(TestSuite) || include("../testsuite/TestSuite.jl")
+using .TestSuite
+
+is_buildkite = get(ENV, "BUILDKITE", "false") == "true"
+
+m = 19
+for T in (BLASFloats..., GenericFloats...)
+    TestSuite.seed_rng!(123)
+    atol = rtol = m * m * TestSuite.precision(T)
+    if !is_buildkite
+        TestSuite.test_mooncake_projections(T, (m, m); atol, rtol)
+        TestSuite.test_mooncake_projections(Diagonal{T, Vector{T}}, (m, m); atol, rtol)
+    end
+end
diff --git a/test/testsuite/TestSuite.jl b/test/testsuite/TestSuite.jl
index 2edd0846..361f125b 100644
--- a/test/testsuite/TestSuite.jl
+++ b/test/testsuite/TestSuite.jl
@@ -116,6 +116,7 @@ include("mooncake/eigh.jl")
 include("mooncake/svd.jl")
 include("mooncake/polar.jl")
 include("mooncake/orthnull.jl")
+include("mooncake/projections.jl")
 
 include("enzyme.jl")
 include("chainrules.jl")
diff --git a/test/testsuite/chainrules.jl b/test/testsuite/chainrules.jl
index bb94664d..52806750 100644
--- a/test/testsuite/chainrules.jl
+++ b/test/testsuite/chainrules.jl
@@ -46,6 +46,7 @@ function test_chainrules(T::Type, sz; kwargs...)
         test_chainrules_svd(T, sz; kwargs...)
         test_chainrules_polar(T, sz; kwargs...)
         test_chainrules_orthnull(T, sz; kwargs...)
+        test_chainrules_projections(T, sz; kwargs...)
     end
 end
 
@@ -587,3 +588,25 @@ function test_chainrules_orthnull(
         )
     end
 end
+
+function test_chainrules_projections(
+        T::Type, sz;
+        atol::Real = 0, rtol::Real = precision(T),
+        kwargs...
+    )
+    summary_str = testargs_summary(T, sz)
+    return @testset "Projections Chainrules AD rules $summary_str" begin
+        A = instantiate_matrix(T, sz)
+        m, n = size(A)
+        if m == n
+            @testset "project_hermitian" begin
+                alg = MatrixAlgebraKit.default_hermitian_algorithm(A)
+                test_rrule(project_hermitian, A, alg; atol, rtol)
+            end
+            @testset "project_antihermitian" begin
+                alg = MatrixAlgebraKit.default_hermitian_algorithm(A)
+                test_rrule(project_antihermitian, A, alg; atol, rtol)
+            end
+        end
+    end
+end
diff --git a/test/testsuite/mooncake/projections.jl b/test/testsuite/mooncake/projections.jl
new file mode 100644
index 00000000..74e22a44
--- /dev/null
+++ b/test/testsuite/mooncake/projections.jl
@@ -0,0 +1,69 @@
+"""
+    test_mooncake_projections(T, sz; kwargs...)
+
+Run all Mooncake AD tests for hermitian and anti-hermitian projections of element type `T`
+and size `sz`.
+"""
+function test_mooncake_projections(T::Type, sz; kwargs...)
+    summary_str = testargs_summary(T, sz)
+    return @testset "Mooncake projection $summary_str" begin
+        test_mooncake_project_hermitian(T, sz; kwargs...)
+        test_mooncake_project_antihermitian(T, sz; kwargs...)
+    end
+end
+
+"""
+    test_mooncake_project_hermitian(T, sz; rng, atol, rtol)
+
+Test the Mooncake reverse-mode AD rule for `project_hermitian` and its in-place variant.
+"""
+function test_mooncake_project_hermitian(
+        T, sz;
+        rng = Random.default_rng(), atol::Real = 0, rtol::Real = precision(T)
+    )
+    return @testset "project_hermitian" begin
+        A = instantiate_matrix(T, sz)
+        B = instantiate_matrix(T, sz)
+        alg = MatrixAlgebraKit.select_algorithm(project_hermitian, A)
+        Mooncake.TestUtils.test_rule(
+            rng, project_hermitian, A, alg;
+            mode = Mooncake.ReverseMode, atol, rtol
+        )
+        Mooncake.TestUtils.test_rule(
+            rng, project_hermitian!, A, A, alg;
+            mode = Mooncake.ReverseMode, atol, rtol
+        )
+        Mooncake.TestUtils.test_rule(
+            rng, project_hermitian!, A, B, alg;
+            mode = Mooncake.ReverseMode, atol, rtol
+        )
+    end
+end
+
+"""
+    test_mooncake_project_antihermitian(T, sz; rng, atol, rtol)
+
+Test the Mooncake reverse-mode AD rule for `project_antihermitian` and its in-place variant.
+"""
+function test_mooncake_project_antihermitian(
+        T, sz;
+        rng = Random.default_rng(), atol::Real = 0, rtol::Real = precision(T)
+    )
+    return @testset "project_antihermitian" begin
+        A = instantiate_matrix(T, sz)
+        B = instantiate_matrix(T, sz)
+        alg = MatrixAlgebraKit.select_algorithm(project_hermitian, A)
+        Mooncake.TestUtils.test_rule(
+            rng, project_antihermitian, A, alg;
+            mode = Mooncake.ReverseMode, atol, rtol
+        )
+        Mooncake.TestUtils.test_rule(
+            rng, project_antihermitian!, A, A, alg;
+            mode = Mooncake.ReverseMode, atol, rtol
+        )
+        Mooncake.TestUtils.test_rule(
+            rng, project_antihermitian!, A, B, alg;
+            mode = Mooncake.ReverseMode, atol, rtol
+        )
+    end
+end
diff --git a/test/testsuite/projections.jl b/test/testsuite/projections.jl
index 3147084a..82091d01 100644
--- a/test/testsuite/projections.jl
+++ b/test/testsuite/projections.jl
@@ -40,6 +40,14 @@ function test_project_antihermitian(
             @test Ba === Ac
             @test isantihermitian(Ba)
             @test Ba ≈ Aa
+
+            # can we supply a destination
+            Ac = deepcopy(A)
+            Ba = instantiate_matrix(T, sz)
+            Ba₂ = project_antihermitian!(Ac, Ba)
+            @test A == Ac
+            @test Ba₂ === Ba
+            @test Ba₂ ≈ Aa
         end
 
         # test approximate error calculation
@@ -79,6 +87,7 @@ function test_project_hermitian(
             Aa = (A - A') / 2
 
             Bh = project_hermitian(A; blocksize = 16)
+
             @test ishermitian(Bh)
             @test Bh ≈ Ah
             @test A == Ac
@@ -91,6 +100,14 @@ function test_project_hermitian(
             @test Bh === Ac
             @test ishermitian(Bh)
             @test Bh ≈ Ah
+
+            # can we supply a destination
+            Ac = deepcopy(A)
+            Bh = instantiate_matrix(T, sz)
+            Bh₂ = project_hermitian!(Ac, Bh)
+            @test A == Ac
+            @test Bh₂ === Bh
+            @test Bh₂ ≈ Ah
         end
 
         # test approximate error calculation