  There has been significant interest in studying the asymptotics of certain
generalised moments, called the moments of moments, of characteristic
polynomials of random Haar-distributed unitary and symplectic matrices, as the
matrix size $N$ goes to infinity. These quantities depend on two parameters $k$
and $q$ and when both of them are positive integers it has been shown that
these moments are in fact polynomials in the matrix size $N$. In this paper we
classify the integer roots of these polynomials and moreover prove that the
polynomials themselves satisfy a certain symmetry property. This confirms some
predictions from the thesis of Bailey. The proof uses the Ehrhart-Macdonald
reciprocity for rational convex polytopes and certain bijections between
lattice points in some polytopes.
