The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of the (smooth) topological generalizations of toric symplectic manifolds and projective toric varieties. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, but this degeneracy is enough to allow for non-simply-connected and non-orientable manifolds, which are excluded from  the topological generalizations mentioned above. In this talk we examine how the topology of an (orientable) toric origami manifold, in particular its fundamental group, can be read from the polytope-like object that represents its orbit space. We conjecture that these results hold for the appropriate topological generalization of the class of toric origami manifolds. These results are from ongoing joint work with Tara Holm.