Toric symplectic manifolds can classified by their moment polytope, and their topology (ordinary and equivariant cohomology) can be read directly from the polytope. A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In this talk we examine the toric origami case: we will describe how toric origami manifolds can also be classified by their combinatorial moment data. We will then explore some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds. We will include a number of geometric and combinatorial examples