The MBE growth of HgTe on CdTe is examined over a hundred-fold range in Hg/Te2 flux ratio and over four separate substrate surface orientations [(111)Te, (111)Te-4 degrees, (112)Te and (001)]. The 77K Hall mobility of the (112)Te and (001) oriented HgTe layers approaches the best bulk values reported to date, although our (111)Te and (111)Te-4 deg oriented HgTe films yield much lower values. The growth process is shown to be very far from thermodynamic equilibrium at our optimal substrate temperature and calculated equivalent Hg beam pressure. Important clues to help understand the kinetics governing the HgTe growth process are uncovered by studying the defects that form under Hg- or Te-rich conditions with cross-sectional transmission electron microscopy (XTEM). Since multilayered structures are an important application for MBE growth of Hg-based semiconductors, we have also examined the interfacial roughness present in HgTe-CdTe superlattices (SL) as a function of growth orientation.