For pt.II see ibid., vol.15, p.337 (1985). Band-structure parameters derived for Pb_1-xSn_xTe in paper I of this series from a Shubnikov-de Haas study are shown to give good agreement with the carrier concentration dependence observed for the anisotropy and the transverse effective mass in p-type PbTe. The parameters are also capable of yielding the x dependence of the anisotropy of the Fermi surfaces in Pb_1-xSn_xTe in the cubic phase at low temperatures. With regard to the band structures deduced for the rhombohedral phase of Pb_1-xSn_xTe, these are found to be capable of explaining the hitherto unexplained de Hass-van Alphen measurements of SnTe at low carrier concentration. The complexity of the Fermi surface for such low-carrier concentration SnTe is discussed.