  Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or
connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal
basis consisting of exponential functions. A result due to Kolountzakis and
Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the
total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal
is pointing at a given direction, must coincide with the total area of those
$(d-1)$-dimensional faces on which the outward normal is pointing at the
opposite direction. In this paper, we prove an extension of this result to
faces of all dimensions between $1$ and $d-1$. As a consequence we obtain that
any spectral polytope $A$ can be dissected into a finite number of smaller
polytopes, which can be rearranged using translations to form a cube.
