  We extend the formalism based on perturbative QCD that was developed in our
previous work, and compute the hyperfine splittings of the bottomonium spectrum
as well as the fine and hyperfine splittings of the charmonium spectrum. All
the corrections up to O(alpha_s^5 m) are included in the computations. We find
agreement (with respect to theoretical uncertainties) with the experimental
values whenever available and give predictions for not yet observed splittings.
