Manifold submetries are a geometric generalization of isometric group actions, as well as of singular Riemannian foliations. For the round sphere, we show that manifold submetries are in one-one correspondence with a certain class of algebras of polynomials intimately related to the Laplace operator, and which generalize the algebra of invariants in classical Invariant Theory. In current work in progress, we use tools from Spectral Geometry to relate this algebra to the geometry of the orbit/leaf space.