  We study the breaking of rotational symmetry on the lattice for irreducible
tensor operators and practical methods for suppressing this breaking. We
illustrate the features of the general problem using an $\alpha$ cluster model
for $^{8}$Be. We focus on the lowest states with non-zero angular momentum and
examine the matrix elements of multipole moment operators. We show that the
physical reduced matrix element is well reproduced by averaging over all
possible orientations of the quantum state, and this is expressed as a sum of
matrix elements weighted by the corresponding Clebsch-Gordan coefficients. For
our $\alpha$ cluster model we find that the effects of rotational symmetry
breaking can be largely eliminated for lattice spacings of $a\leq 1.7$ fm, and
we expect similar improvement for actual lattice Monte Carlo calculations.
