  We investigate arithmetic, geometric and combinatorial properties of
symmetric edge polytopes. We give a complete combinatorial description of their
facets. By combining Gr\"obner basis techniques, half-open decompositions and
methods for interlacing polynomials we provide an explicit formula for the
$h^\ast$-polynomial in case of complete bipartite graphs. In particular, we
show that the $h^\ast$-polynomial is $\gamma$-positive and real-rooted. This
proves Gal's conjecture for arbitrary flag unimodular triangulations in this
case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).
