Abstract The dominant features of the band structure have been analysed by studying the fundamental reflectivity over the energy range 0.1–12 eV and the absorption over the range 0.1–1.5 eV. Using linearly polarized light, results have been obtained for different orientations of the plane of polarization relative to the three-fold axis in order that the selection rules for optical transitions may be taken into account. The probable symmetries of the top valence band and lowest conduction band have been established at several points in the Brillouin zone. The band structure of bismuth telluride and bismuth selenide appear to be rather similar apart from a consistent energy shift; the optical gap of bismuth selenide however, is lower than previously published values. Like the optical energy gap a number of low energy maxima in the joint density of states appear to arise from transitions along the Γ-A or Γ-D directions of the Brillouin zone. The dominant inter-band transitions above 1.5 eV seem to be due to critical points at or between the Γ and Z points. Estimates of 1.1 eV and 1.4 eV have been obtained for the spin orbit splitting of the respective valence and conduction bands involved in those transitions in bismuth telluride.