 Symmetric product criterion for symmetric matrices

If $A^T=A$ and $B^T=B$, then $(AB{)}^T=B^T{A}^T=BA$. Hence their product is a symmetric matrix exactly when they commute. For example, $A=\mathrm{diag}(1,0)$ and $B=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ are symmetric but have nonsymmetric product $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$. This criterion differs from asking whether an arbitrary matrix can be factored as a product of symmetric matrices.