  Standard toric geometry methods used to construct Calabi-Yau varieties may be
extended to complete intersections in non-Fano varieties encoded by star
triangulating non-convex polytopes. Similarly, mirror symmetry is conjectured
to hold in terms of a transpolar duality generalizing the original construction
of Batyrev and Borisov. The associated mirror pairs naturally include certain
flip-folded, multi-layered multihedral objects, inclusively named VEX
multitopes, and a correspondingly generalized transpolar duality. These
self-overlaying VEX multitopes, long since known in pre-symplectic geometry,
are found to correspond to certain non-algebraic but smooth toric spaces with
Chern classes that satisfy the standard Todd-Hirzebruch identities. The
computation of diffeomorphism invariants, including characteristic submanifold
intersection numbers, corroborates their recent inclusion in the connected web
of Calabi-Yau spaces and associated string compactifications: They arise
together with the standard (Fano/reflexive polytope) constructions, within
deformation families of generalized complete intersections in products of
projective spaces.
