We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical $4$-polytope that has a nonorientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995).
¶ More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into {\small $\R^3$} is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubification of Boy's surface as a dual manifold immersion, and with an odd number of facets. Our explicit example has 17,718 vertices and 16,533 facets. Thus we get a parity-changing operation for three-dimensional cubical complexes (hex meshes); this solves problems of Eppstein, Thurston, and others.