Recall that given a nondegenerate polytope $P \subset \mathbb{R}^n$ which is the convex set of some vectors with integral coordinates, the Erhart polynomial $p_P(t)$ a polynomial such that $p_P(t)$ for natural numbers $t$ is the number of integral lattice points in the $t$-fold dilation $tP$ of $P$.

Do there exist two polytopes $P, P'$ which have equal Erhart polynomials, are preserved under Euclidean reflections along all coordinate hyperplanes $\{x_i = 0\} \subset \mathbb{R}^n$, and are not $GL(n, \mathbb{Z})$ equivalent, i.e. there is no integral matrix $A$ with $det(A) = \pm 1$ such that $AP = P'$?

In fact I would be happy if there were two polytopes which are convex hulls of rational vectors and are symmetric under reflection across all coordinate hyperplanes, which are not $GL(n, \mathbb{Z})$ equivalent but have the same Erhart quasi-polynomials (these are certain functions characterized in the same way as the Erhart polynomials using the same definition of their values on the positive natural numbers; they just cease to be polynomial functions in the rational case.)