The molecular structures and energetic stabilities of the three pure polymorphic forms of crystalline HMX were calculated using a first-principles electronic-structure method. The computations were performed using the local density approximation in conjunction with localized "fireball" orbitals and a minimal basis set. Optimized cell parameters and molecular geometries were obtained, subject only to preservation of the experimental lattice angles and relative lattice lengths. The latter constraint was removed in some calculations for beta-HMX. Within these constraints, the comparison between theory and experiment is found to be good. The structures, relative energies of the polymorphs, and bulk moduli are in general agreement with the available experimental data.