diff --git a/src/diagonal.jl b/src/diagonal.jl
index e60fb009..63cff1c8 100644
--- a/src/diagonal.jl
+++ b/src/diagonal.jl
@@ -970,6 +970,10 @@ function inv(D::Diagonal{T}) where T
     Diagonal(Di)
 end
 
+# Ensure doubly wrapped matrices use efficient diagonal methods and return a Symmetric/Hermitian type
+inv(A::Symmetric{<:Number,<:Diagonal}) = Symmetric(inv(A.data), sym_uplo(A.uplo))
+inv(A::Hermitian{<:Number,<:Diagonal}) = Hermitian(inv(real(A.data)), sym_uplo(A.uplo))
+
 function pinv(D::Diagonal{T}) where T
     Di = similar(D.diag, typeof(inv(oneunit(T))))
     for i = 1:length(D.diag)
diff --git a/test/symmetric.jl b/test/symmetric.jl
index 707b392d..ae10618f 100644
--- a/test/symmetric.jl
+++ b/test/symmetric.jl
@@ -874,6 +874,21 @@ end
     @test det(Hermitian(A))::Float64 == det(A) == 0.0
 end
 
+@testset "issue #1437: inverse of Symmetric|Hermitian{<:Any,<:Diagonal} returns of Symmetric|Hermitian{<:Any,<:Diagonal}" begin
+    Dreal    = Diagonal(randn(3))
+    Dcomplex = Diagonal(randn(ComplexF64, 3))
+    # without wrapper
+    invDreal = inv(Dreal)
+    invDcomplex = inv(real(Dcomplex)) # because Hermitian implies a real diagonal
+    # with wrapper
+    SDreal = Symmetric(Dreal)
+    HDcomplex = Hermitian(Dcomplex)
+    @test inv(SDreal)::Symmetric{Float64,typeof(Dreal)} ≈ invDreal
+    @test inv(HDcomplex)::Hermitian{Float64,typeof(Dreal)} ≈ invDcomplex
+    Dcomplex[2,2] = 0
+    @test_throws SingularException inv(HDcomplex)
+end
+
 @testset "symmetric()/hermitian() for Numbers" begin
     @test LinearAlgebra.symmetric(1) == LinearAlgebra.symmetric(1, :U) == 1
     @test LinearAlgebra.symmetric_type(Int) == Int