The breakdown of Friedel's law has been observed in backscatter Kikuchi diffraction patterns (BKDP) obtained in the scanning electron microscope (SEM) from a series of zincblende structures including GaAs, InP, GaSb, CdHgTe and the minerals sphalerite (ZnS), chalcopyrite (CuFeS2) and tetrahedrite (Cu12Sb4S13). Differences in intensities were observed between the reflections 51BAR1 and 5BAR1BAR1 in InP, GaSb, CdHgTe and sphalerite, thus allowing the non-centrosymmetric point group BAR43m to be determined. In GaAs, differences in intensities were noted between BAR511 and BAR5BAR11. In chalcopyrite and tetrahedrite, non-equivalent intensities were observed between BAR215 and 2BAR1BAR5 and between 3BAR1BAR2 and 31BAR2, respectively. In addition, BKDPs obtained from chalcopyrite revealed a small displacement at the point where the pair of equivalent reflections BAR406 and 460 intersect within the Kikuchi band 02BAR2. The presence of this displacement together with observation of the breakdown of Friedel's law confirmed the tetragonal point group BAR42m for chalcopyrite. Although the point groups of GaAs, chalcopyrite and tetrahedrite were derived successfully using BKDPs, determination of their space groups proved unsuccessful. The superstructure reflections were invisible because the structure factors are very small. The behaviour of the invisible 200 reflection in GaAs is investigated using many-beam dynamical intensity profiles calculated across the h00 systematic row of reflections. Dynamical intensity profiles calculated across the h00 systematic rows of reflections for Ge, InP and sphalerite are also discussed.