  We introduce a classification of the radial spin textures in momentum space
that emerge at high-symmetry points in crystals characterized by non-polar
chiral point groups ($D_2$, $D_3$, $D_4$, $D_6$, $T$, $O$). 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
non-polar 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 $\mathrm{Ta}_2
\mathrm{Se}_8\mathrm{I}$, the material proposed for dark matter detection
$\mathrm{Ag}_3\mathrm{Au}\mathrm{Te}_2$ and heazlewoodite
$\mathrm{Ni}_3\mathrm{S}_2$, 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.
