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.