Here, we report a study of the molecular-beam epitaxial (MBE) growth of HgCdTe on (100) and (111)B CdTe substrates. We show that the growth on the (100) orientation is easier to control than on the (111)B orientation, mainly because twinning, caused by excess Hg, is not observed. The structural quality, as probed by double-crystal x-ray rocking curve, is consistently of better quality for the (100) orientation. As-grown n- and p-type HgCdTe epilayers grown on both orientations exhibit high Hall mobilities. For the first time, p-type HgCdTe grown by MBE on the (100) orientation is demonstrated in this work. The HgCdTe layers grown on the (100) orientation have normal n- and p-type annealing behavior, but those grown on the (111)B behave abnormally, probably because of the presence of twins and stacking faults like defects created during growth. Improving MBE growth conditions could lead to a reduction in density of these defects on the (111)B plane.