  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 symplectic manifolds, studied by Cannas da Silva,
Guillemin and Pires, who classified toric origami manifolds by combinatorial
origami templates. In this paper, we examine the topology of toric origami
manifolds that have acyclic origami template and co-orientable folding
hypersurface. We prove that the cohomology is concentrated in even degrees, and
that the equivariant cohomology satisfies the GKM description. Finally we show
that toric origami manifolds with co-orientable folding hypersurface provide a
class of examples of Masuda and Panov's torus manifolds.
