For testing a probability distribution on a compact Riemannian manifold for symmetry under the action of a given group of isometries, two classes of invariant tests are proposed and some properties noted. These tests are based on Sobolev norms and generalize Gine's Sobolev tests of uniformity. For general compact manifolds randomization tests analogous to Wellner's tests for the two-sample case are suggested. For the circle, distribution-free tests of symmetry based on uniform scores are provided.
"Sobolev Tests for Symmetry of Directional Data." Ann. Statist. 11 (4) 1225 - 1231, December, 1983. https://doi.org/10.1214/aos/1176346335