Is there a polyhedral version of sphere eversion (Smale's theorem), where you take a polyhedron with triangular faces homeomorphic to the sphere, and continuously move the vertices such that no dihedral angle becomes zero (that is, no pair adjacent faces ever overlap) and such that it ends inside-out of how it started?
If so, how many vertices do we need? A simple argument shows the tetrahedron cannot be everted in that manner. I don't think the octahedron can, either. I imagine this probably needs a large polyhedron to work.
(Alternate question: What if you require that the dihedral angles never leave $(\pi-\epsilon,\pi+\epsilon)$ for some $\epsilon$?)
Apery and Denner showed that the cuboctahedron is the minimum polyhedron that can be everted in the way you describe.