Let $C$ be an idempotent and symmetric matrix and its spectral composition is given by $$C = ADA^T$$ I cannot see how you can rewrite this to be $$D = A^TCA$$
I've found this reformulation in a book without explanation, but I cannot see why this should be correct and how you could get there. So any hints would be highly appreciated.
Just for the sake of having an answer:
$C = A D A^{T} \Rightarrow A^{-1} C=A^{-1} A DA^{T} \Rightarrow A^{T}C= I D A^{T} \Rightarrow A^{T} C = D A^{T} \Rightarrow A^{T} C A = D A^{T}A \Rightarrow A^{T} C A = D I \Rightarrow D = A^{T} C A.$
As @copper.hat mentioned $A^{-1} = A^{T}$, since the matrix $A$ is orthogonal.