Everybody knows that a square matrix $A$ has the same eigenvalues as $A^T$. And it is clear that if $A^T=BAB^{-1}$ then $B$ maps eigenvectors of $A$ to those of $A^T$. But I have not found any discussion of the benefits of knowing $B$. Perhaps it is unusual to know $B$ exactly, without having analyzed $A$ completely. But I have some examples where this is known.

It seems at least to be nontrivial information. For example, with $A$ real and $B$ real and symmetric, you know an indefinite inner product $\langle x,y\rangle = x^HBy$ with respect to which $A$ is symmetric. This gives certain orthogonality relations (for example non-real eigenvalues have null eigenvectors) but it doesn't seem to lead to anything quantitative about eigenvalues.

The worst case may be for symmetric $A$, $A^T=IAI^{-1}$ tells you nothing.

So the question is

Have examples been studied, where knowing $B$ in $A^T=BAB^{-1}$ helped to find eigenvalues of $A$?