The electronic structure and vibrational properties of the four polymorphs of crystalline octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) have been studied using density functional theory within the local density approximation. The results show that the states of N in the ring make more important contributions to the valence bands than these of C and N of NO2 and so N in the ring acts as an active center. From the low frequency to high-frequency region, the molecular motions of the vibrational frequencies for the four HMX polymorphs present unique features. It is also noted that there is a relationship between the band gap and impact sensitivity for the four HMX polymorphs. From the cell bond order per unit volume, we may infer the variation order of crystal bonding for the four polymorphs and so predict their impact sensitivity order as follows: beta-HMX < gamma-HMX < alpha-HMX < delta-HMX, which is in agreement with their experimental order.