We present a systematic analysis of the intensity anisotropy of fourth harmonic generation (FHG) in reflection from centrosymmetric crystals of classes m3m (e.g. diamond structure) and 6/mmm (e.g. graphite) and non-centrosymmetric crystals of class <(4)over bar 3m> (e.g. zincblende structure). Surface dipole FH contributions are analyzed for six symmetry classes (2mm, 4mm, 3mm, m, 6mm, 5-fold) which encompass the (001), (111), and (110) surfaces of the diamond-zincblende crystals, the (001) surface of graphite. and the (00001) surface of an icosahedral quasicrystal.