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 non-orientable 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 R3 is PL-equivalent to a dual manifold immersion of a cubical
4-polytope. As an instance we obtain a cubical 4-polytope with a cubation 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 3-dimensional
cubical complexes (hexa meshes); this solves problems of Eppstein, Thurston, and others.