Ga and In crystallize in unusual open ground-state crystal structures. Recent experiments have discovered that Ga under high pressure transforms into a close-packed structure, while this has so far not been observed for In. Results from first principles calculations explain in a simple way this difference in behavior. We predict a so far undiscovered transition of In to a close-packed structure at extreme pressures and show that the structure determining mechanism originates from the degree of s-p mixing of the valence orbitals. Group-III elements are shown to strongly disobey the standard corresponding-state rule.