Equivalent spherical harmonics

It is known that the 2l+1 spherical harmonics Y_lm can be transformed into cyclically equivalent orbitals, of which only a few examples have so far been given explicitly. In this article the totality of such cyclic sets is derived. It is demonstrated that other kinds of equivalent spherical harmonics do not exist. Finally a set of five equivalent d orbitals related to icosahedral symmetry is introduced.