More than six decades have passed since Wigner and Huntington(1) proposed that hydrogen might form a solid metallic phase at high density with characteristics similar to the alkali metals, This possibility has been investigated using the diamond-anvil cell to compress the crystalline state of molecular hydrogen(2), but there is still no definitive evidence for a dense, low-temperature metallic state, Below 140 K, solid hydrogen undergoes a transition at about 1.5 million atmospheres between two orientationally ordered states, The intermolecular vibrational mode (the vibron) shifts to a lower frequency at this transition(3,4), and becomes strongly infrared-active(5). So far as is known, hydrogen remains in this phase to the highest pressures yet reached. Here we report first-principles calculations of the structure of this phase using electronic density-functional theory. We find that it develops a spontaneous polarization at around ninefold compression relative to the volume at 1 atmosphere and that there is a corresponding movement of proton pairs away from their ideal lattice sites. Such behaviour can explain why the vibron becomes infrared-active, ie, and rationalizes the direction and mass-dependence (in experiments on deuterium) of the shift of the vibron frequency. In the polarized state, the previously decreasing bandgap widens again, and so its appearance might delay the transition to the elusive metallic state.