linear algebra - matrices self-adjoint with respect to some inner product

Is it possible to give a nice characterization of matrices $A in R^{n times n}$ which are self-adjoint with respect to some inner product?

These matrices include all symmetric matrices (of course) and some nonsymmetric ones: for example, the transition matrix of any (irreducible) reversible Markov chain will have this property.

Naturally, all such matrices must have real eigenvalues, though I do not expect that this is a sufficient condition (is it?).

About the only observation I have is that since any inner product
can be represented as $langle x,y rangle = x^T M y$ for some positive definite matrix $M$, we are looking for matrices $A$ which satisfy $A^T M = M A$ or $M^{-1} A^T M = A$. In other words, we are looking for real matrices similar to their transpose with a positive definite similarity matrix.