  We study lattice path matroid polytopes using their alcoved triangulation. We
characterize Gorenstein lattice path matroid polytopes, yielding a new class of
matroids satisfying the unimodality conjecture of de Loera, Haws, and
K{\"o}ppe. Further, we characterize matroids whose polytopes are order
polytopes as a special class of lattice path matroids, called snakes. Finally,
we give combinatorial interpretations of the volumes and $h^*$-vectors of
lattice path matroids of rank $2$ based on their snake decomposition.
