This turns out to be more complicated that I first thought it'd be. Apparently the graphs you are asking about are usually called normal digraphs and a proper characterization does not seem to be known. This recent paper treats characterization in the special case of Cayley digraphs and also refers to previous work on other cases (alas, almost all of it is in not-immediately-accessible-online places).

There is case, I think, that is easy to work out: graph where in-degrees equals\ the out-degrees. The write-up here indicates (once again, based on a 2005 paper I can't access here and now) that such graphs (called balanced) have a normal Laplacian matrix, which is easily seen to be equivalent to having a normal adjacency matrix.

This turns out to be more complicated that I first thought it'd be. Apparently the graphs you are asking about are usually called normal digraphs and a proper characterization does not seem to be known. This recent paper treats characterization in the special case of Cayley digraphs and also refers to previous work on other cases (alas, almost all of it is in not-immediately-accessible-online places).

There is case, I think, that is easy to work out: graph where in-degrees equals\ the out-degrees. The write-up here indicates (once again, based on a 2005 paper I can't access here and now) that such graphs (called balanced) have a normal Laplacian matrix, which is easily seen to be equivalent to having a normal adjacency matrix.