diff --git a/ext/StaticArraysChainRulesCoreExt.jl b/ext/StaticArraysChainRulesCoreExt.jl
index 5f7904aa..3ae86a5c 100644
--- a/ext/StaticArraysChainRulesCoreExt.jl
+++ b/ext/StaticArraysChainRulesCoreExt.jl
@@ -15,12 +15,14 @@ end
 
 # Project SArray to SArray
 function ProjectTo(x::SArray{S, T}) where {S, T}
-    return ProjectTo{SArray}(; element = CRC._eltype_projectto(T), axes = Size(x))
+    # We have a axes field because it is expected by other ProjectTo's like the one for Transpose
+    return ProjectTo{SArray}(; element = CRC._eltype_projectto(T), axes = axes(x),
+        size = Size(x))
 end
 
 @inline _sarray_from_array(::Size{T}, dx::AbstractArray) where {T} = SArray{Tuple{T...}}(dx)
 
-(project::ProjectTo{SArray})(dx::AbstractArray) = _sarray_from_array(project.axes, dx)
+(project::ProjectTo{SArray})(dx::AbstractArray) = _sarray_from_array(project.size, dx)
 
 # Adjoint for SArray constructor
 function rrule(::Type{T}, x::Tuple) where {T <: SArray}
diff --git a/test/chainrules.jl b/test/chainrules.jl
index 7dbf8e8a..c586432a 100644
--- a/test/chainrules.jl
+++ b/test/chainrules.jl
@@ -1,7 +1,24 @@
-using StaticArrays, ChainRulesCore, ChainRulesTestUtils, JLArrays, Test
+using StaticArrays, ChainRulesCore, ChainRulesTestUtils, JLArrays, LinearAlgebra, Test
 
-@testset "Chain Rules Integration" begin
+@testset "ChainRules Integration" begin
     @testset "Projection" begin
+        # There is no code for this, but when argument isa StaticArray, axes(x) === axes(dx)
+        # implies a check, and reshape will wrap a Vector into a static SizedVector:
+        pstat = ProjectTo(SA[1, 2, 3])
+        @test axes(pstat(rand(3))) === (SOneTo(3),)
+
+        # This recurses into structured arrays:
+        pst = ProjectTo(transpose(SA[1, 2, 3]))
+        @test axes(pst(rand(1,3))) === (SOneTo(1), SOneTo(3))
+        @test pst(rand(1,3)) isa Transpose
+
+        # When the argument is an ordinary Array, static gradients are allowed to pass,
+        # like FillArrays. Collecting to an Array would cost a copy.
+        pvec3 = ProjectTo([1, 2, 3])
+        @test pvec3(SA[1, 2, 3]) isa StaticArray
+    end
+
+    @testset "Constructor rrules" begin
         test_rrule(SMatrix{1, 4}, (1.0, 1.0, 1.0, 1.0))
         test_rrule(SMatrix{4, 1}, (1.0, 1.0, 1.0, 1.0))
         test_rrule(SMatrix{2, 2}, (1.0, 1.0, 1.0, 1.0))