We introduce a classification of the radial spin textures in momentum space that emerge at the high-symmetry points in crystals characterized by nonpolar chiral point groups (D2, D3, D4, D6, T, O). Based on the symmetry constraints imposed by these point groups in a vector field, we study the general expression for the radial spin textures up to third order in momentum. Furthermore, we determine the high-symmetry points of the 45 nonpolar chiral space groups supporting a radial spin texture. These two principles are used to screen materials databases for archetypes that go beyond the basic hedgehog radial spin texture. Among the selected materials we highlight the axion insulator candidate Ta2Se8I, the material proposed for dark matter detection Ag3AuTe2, and heazlewoodite Ni3S2, a conventional metal predicted to exhibit current-induced spin polarization. We point out that the symmetry analysis proposed in this Letter is more general and extends to studying other vector properties in momentum space.