I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if they lie on a common edge in $P$.
The orthogonal symmetry group of $P$ induces a permutation symmetry group on its graph. We say that a polytope is vertex-, edge-, arc- and/or half-transitive if the symmetry group induced on its graph acts in the respective way.
Question: Are there half-transitive polytopes, i.e. polytopes which are vertex- and edge-transitive, but not arc-transitive?
I looked a bit into chiral polytopes (i.e. two flag orbits), but I read somewhere that these only exist for abstract polytopes, and not for convex ones. However, I think that chirality is a much stronger requirement for a polytope than half-transitivity (as the latter only speaks about vertices and edges instead of flags).