  Manifold submetries of the round sphere are a class of partitions of the
round sphere that generalizes both singular Riemannian foliations, and the
orbit decompositions by the orthogonal representations of compact groups. We
exhibit a one-to-one correspondence between such manifold submetries and
maximal Laplacian algebras, thus solving the Inverse Invariant Theory problem
for this class of partitions. Moreover, a solution to the analogous problem is
provided for two smaller classes, namely orthogonal representations of finite
groups, and transnormal systems with closed leaves.
