Medium-energy ion scattering measurements and Monte Carlo computer simulations have been used to study the melting behaviour of the two open low-index surfaces of indium: In(110) and In(011). Because of the tetragonal lattice structure, the atomic density at the In(110) surface is 3.5% lower than that at the In(011) surface. The corresponding higher surface free energy of the (110) surface is expected to give rise to a slightly stronger surface melting effect at this face than at the (011) face. The ion-scattering measurements indeed show such a difference in melted-layer thickness at temperatures close to the bulk melting point, Tm=429.76 K. Surprisingly, however, the order of the onset of surface disordering is reversed. In(110) starts disordering about 25 K closer to Tm=429.76 than In(011). We attribute this counter-intuitive difference in disordering onset to the difference in adatom-vacancy creation energies at the two surfaces. This idea is corroborated by energy calculations and Monte Carlo computer simulations, in which a Finnis-Sinclair interaction potential between indium atoms was employed. The slightly increased nearest-neighbour distance on the more densely-packed In(011) surface leads to a decreased creation energy of adatoms and vacancies. The resulting higher densities of adatoms and vacancies make In(011) unstable at a lower temperature than In(110). These results strongly suggest that the onset of surface disordering involves a mechanical surface instability.