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 3a113c20..c51184ae 100644
--- a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
+++ b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
@@ -779,4 +779,49 @@ 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)
+            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/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/testsuite/chainrules.jl b/test/testsuite/chainrules.jl
index b4126c59..f937e0b0 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
 
@@ -610,3 +611,25 @@ 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.jl b/test/testsuite/mooncake.jl
index 29d65e31..9ca869d2 100644
--- a/test/testsuite/mooncake.jl
+++ b/test/testsuite/mooncake.jl
@@ -229,6 +229,7 @@ function test_mooncake(T::Type, sz; kwargs...)
         if T <: Number
             test_mooncake_orthnull(T, sz; kwargs...)
         end
+        test_mooncake_projections(T, sz; kwargs...)
     end
 end
 
@@ -537,3 +538,29 @@ function test_mooncake_orthnull(
         test_pullbacks_match(((X, Nᴴ) -> right_null!(X, Nᴴ; alg = :lq)), right_null_lq, A, Nᴴ, ΔNᴴ)
     end
 end
+
+function test_mooncake_projections(
+        T::Type, sz;
+        atol::Real = 0, rtol::Real = precision(T),
+        kwargs...
+    )
+    summary_str = testargs_summary(T, sz)
+    return @testset "Projections Mooncake AD rules $summary_str" begin
+        A = instantiate_matrix(T, sz)
+        m, n = size(A)
+        if m == n
+            @testset "project_hermitian" begin
+                Aₕ = project_hermitian(A)
+                ΔAₕ = make_mooncake_tangent(Aₕ)
+                Mooncake.TestUtils.test_rule(rng, project_hermitian, A; is_primitive = false, mode = Mooncake.ReverseMode, atol, rtol)
+                test_pullbacks_match(project_hermitian!, project_hermitian, A, Aₕ, ΔAₕ)
+            end
+            @testset "project_antihermitian" begin
+                Aₐ = project_antihermitian(A)
+                ΔAₐ = make_mooncake_tangent(Aₐ)
+                Mooncake.TestUtils.test_rule(rng, project_antihermitian, A; is_primitive = false, mode = Mooncake.ReverseMode, atol, rtol)
+                test_pullbacks_match(project_antihermitian!, project_antihermitian, A, Aₐ, ΔAₐ)
+            end
+        end
+    end
+end