I'm not sure this is generically true, but as part of something I'd like to prove I'm hoping to make use of something along the lines of being able to write for some symmetric, nonzero $B$
$$ B = A B A^\intercal $$
and using that to infer that $A$ must be the identity matrix
I think I can infer that $A$ is a similarity transform, and therefore unitary, and I know that I can write
$$ B = (A^n) B (A^\intercal)^n $$
for any positive integer $n$, but beyond that all I know is that I'm trying to find the centralizers of this matrix in the group of unitary matrices