The electric quadrupole-quadrupole ($\mathcal{E}_{qq}$) interaction is believed to play an important role in the broken symmetry transition from Phase I to II in solid hydrogen. To evaluate this, we study structures adopted by purely classical quadrupoles using Markov Chain Monte Carlo simulations of fcc and hcp quadrupolar lattices. Both undergo first-order phase transitions from rotationally ordered to disordered structures, as indicated by a discontinuity in both quadrupole interaction energy ($\mathcal{E}_{qq}$) and its heat capacity. Cooling fcc reliably induced a transition to the P$a3$ structure, whereas cooling hcp gave inconsistent, frustrated and $c/a$-ratio-dependent broken symmetry states. Analysing the lowest-energy hcp states using simulated annealing, we found P$6_3/m$ and P$ca2_1$ structures found previously as minimum-energy structures in full electronic structure calculations. The candidate structures for hydrogen Phases III-V were not observed. This demonstrates that $\mathcal{E}_{qq}$ is the dominant interaction determining the symmetry breaking in Phase II. The disorder transition occurs at significantly lower temperature in hcp than fcc, showing that the $\mathcal{E}_{qq}$ cannot be responsible for hydrogen Phase II being based on hcp.