We use first-principles calculations to investigate the band structure evolution of WX2/MoX2 (X = S, Se) heterobilayers under a perpendicular electric field. We characterize the extent to which the type II band alignment in these compounds can be tuned or inverted electrostatically. Our results demonstrate two effects of the stacking configuration. First, different stackings produce different net dipole moments, resulting in band offset variations that are larger than 0.1 eV. Second, based on symmetry constraints that depend on stacking, a perpendicular electric field may hybridizeWX(2) and MoX2 bands that cross at the Brillouin zone corner K. Our results suggest that external electric fields can be used to tune the physics of intralayer and interlayer excitons in heterobilayers of transition metal dichalcogenides.