The contact resistance between two dissimilar semiconductors is determined by the carrier transmission through their interface. Despite the ubiquitous presence of interfaces, quantitative simulation of charge transport across such interfaces is difficult, limiting the understanding of interfacial charge transport. This work employs Green's functions to study the charge transport across representative Si/Ge interfaces. For perfect interfaces, it is found that the transmittance is small and the contact resistance is high, not only because the mismatch of carrier pockets makes it hard to meet the momentum conservation requirement, but also because of the incompatible symmetries of the Bloch wave functions of the two sides. In contrast, atomic mixing at the interface increases the carrier transmittance as the interface roughness opens many nonspecular transmission channels, which greatly reduces the contact resistance compared with the perfect interface. Specifically, we show that disordered interfaces with certain symmetries create more nonspecular transmission. The insights from our study will benefit the future design of high-performance heterostructures with low contact resistance.