(This characterisation is already given here; It is added as cw-answer for better visibility).

Quote from the paper Jørgensen, [Normally Regular Digraphs](https://arxiv.org/pdf/1410.8424.pdf) (2014)

> (The $(i, j)$ entry of $AA^t$ (respectively $A^tA$) is the number of common out-neighbours (respectively in-neighbours) of $x_i$ and $x_j$).  

>We say that a digraph is normal if its adjacency matrix $A$ is normal,
i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for
any two (not necessarily distinct) vertices $x$ and $y$, the number of common
out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours
of $x$ and $y$.