The spacings between bound-state levels of the Schrödinger equation with the same principal quantum number N but orbital angular momenta \ell differing by unity are found to be nearly equal for a wide range of power potentials V = \lambda r^\nu, with E_{N \ell} \approx F(\nu, N) - G(\nu,N) \ell. Semiclassical approximations are in accord with this behavior. The result is applied to estimates of masses for quarkonium levels which have not yet been observed, including the 2P c \bar c states and the 1D b \bar b states.