Let $A$ and $B$ be two real symmetric matrices. If $AB=(BA)^{T}$ then, is there any relation between the eigenvalues of $A+B$ and eigenvalues of $A$, $ B$?

If $A$ and $B$ be two real symmetric matrices such that $AB=BA$, then the eigenvalues of $A+B$ is the sum of eigenvalues of $A$ and $B$ in some order. The similar way, can we express the eigenvalues $A+B$ using the eigenvalues of $A$ and $B$ when $AB$=transpose of $BA$.