  We initiate the study of a type $C_n$ generalization of the lattice path
matroids defined by Bonin, de Mier, and Noy. These are delta matroids whose
feasible sets are in bijection with lattice paths which are symmetric along the
main diagonal. We describe deletion, contraction, enveloping matroids, and the
homogeneous components of these delta matroids in terms of the combinatorics of
symmetric lattice paths.
  In the second half, we study the convex geometry of the feasible polytopes of
lattice path delta matroids. We show that these polytopes decompose into the
feasible polytope of special lattice paths corresponding to intervals in the
type $C_n$ Gale order whose related Richardson variety is a toric variety.
Further, these special polytopes inherit a unimodular triangulation from
Stanley's Eulerian triangulation of the hypercube. As a consequence, the volume
of these polytopes is the proportion of permutations with specific ascent sets.

Combinatorics