Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group.Consider the set [Hscr]P(k) of all continuous functions on Rn satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set [Hscr]G of all G-harmonic functions. Then a necessary and sufficient condition for the equality [Hscr]P(k)=[Hscr]G is given in terms of a distinguished invariant basis, called the canonical invariant basis, of G.