The eigenvalues of a Hubbard Hamiltonian for a four-center tetrahedral cluster are calculated exactly. Full use is made of the symmetry of the problem, which is analyzed for an arbitary number of electrons, 0N8. Comparison is made with the phenomenological Hund's-rule predictions for the ground states. The diversity of the low-energy states is surprising: magnetic and nonmagnetic solutions, single and degenerate representations, accidental degeneracies, and symmetry crossovers are all found for the ground states. Implications for three-dimensional lattices are discussed.