  A major theoretical challenge of studying twisted transition metal
dichalcogenide (TMD) bilayers is that the unit cell of such structures is very
large and therefore difficult to address using first-principles methods.
However, twisted TMD bilayers form moir\'e patterns, which consist of regions
of commensurate stacking, either smoothly interpolated into one another or
separated by sharp domain walls. In this paper, we study twisted TMD bilayers
by studying the properties of the constituent commensurate structures. Using
density functional theory (DFT), we compute band structures for
commensurately-stacked MoS$_2$/WS$_2$ and MoSe$_2$/WSe$_2$ bilayers in both
$0^\circ$ and $60^\circ$ orientations, and we highlight variations in band
structures across different commensurate geometries. These band structure
variations arise from diverse factors such as metal atom asymmetry between
layers (Mo vs. W), differences in interlayer hybridization, and Brillouin zone
alignment. We comment on the consequences of such band structure differences
for optical experiments and on the effects of strain on moir\'e pattern
electronic structure.
