  Starting from any finite simple graph, one can build a reflexive polytope
known as a symmetric edge polytope. The first goal of this paper is to show
that symmetric edge polytopes are intrinsically matroidal objects: more
precisely, we prove that two symmetric edge polytopes are unimodularly
equivalent precisely when they share the same graphical matroid. The second
goal is to show that one can construct a generalized symmetric edge polytope
starting from every regular matroid. Just like in the usual case, we are able
to find combinatorial ways to describe the facets and an explicit regular
unimodular triangulation of any such polytope. Finally, we show that the
Ehrhart theory of the polar of a given generalized symmetric edge polytope is
tightly linked to the structure of the lattice of flows of the dual regular
matroid.
