Let's say I have a combinatorially self-dual polytope $P\subseteq\Bbb R^d$, i.e., its face lattice is isomorphic to its dual (you reverse the direction of the lattice order).

> **Question:** Is it always possible to realize $P$ geometrically, so that $P$ and its polar
$$P^\circ := \{x\in\Bbb R^d  \mid \langle x,s\rangle \ge 1 \text{ for all $s\in P$}\}$$
are geometrically related in some sense, e.g. via orthogonal, linear, affine or projective transformations? In other words: so that they are *geometrically self-dual*?

I would prefer to have a realization of the duality under the weakest possible transformations (i.e. orthogonal), but I wonder whether more general are necessary.