  We explore the breaking of rotational symmetry on the lattice for bound state
energies and practical methods for suppressing this breaking. We demonstrate
the general problems associated with lattice discretization errors and
finite-volume errors using an $\alpha$ cluster model for $^8$Be and $^{12}$C.
We consider the two and three $\alpha$-particle systems and focus on the lowest
states with non-zero angular momentum which split into multiplets corresponding
to different irreducible representations of the cubic group. We examine the
dependence of such splittings on the lattice spacing and box size. We find that
lattice spacing errors are closely related to the commensurability of the
lattice with the intrinsic length scales of the system. We also show that
rotational symmetry breaking effects can be significantly reduced by using
improved lattice actions, and that the physical energy levels are accurately
reproduced by the weighted average of a given spin multiplets.
