Do a polytope $P$ and its polar dual $P^\circ$ have the same centroid?

Question

One natural choice for a center point of a convex polytope $P\subset\Bbb R^d$ is the average of all its vertices

$$c(P):=\frac{v_1+\cdots +v_n}{n}.$$

Call $P$ centered if $c(P)=0$

Question: If $P$ is centered, is its polar dual $P^\circ$ centered too?

This is true if $P$ has an irreducible group of symmetries, because $P$ and $P^\circ$ have the same symmetries and $c(P)$ is the only point fixed by these symmetries. This implies a positive answer to this question for simplices.

Answer 1

As it turns out, counterexamples are easy to find.

The trapezoid has its centroid in the origin, but its dual has not. Both facts are easy to argue for by geometric considerations.