I think the important thing here is that we do not need to rotate the polytope to turn it into a lattice polytope. All we need is to parallelly move the supporting hyperplanes of a Delzant polytope since the normal vectors of its facets are all integral. Then the diffeomorphism type of the corresponding symplectic manifold is preserved. For an arbitrary simple convex polytope, we need to use both rotations and translations of its supporting hyperplanes to turn it into a lattice polytope.