  We calculate the masses and leptonic decay widths of the bottomonium $b\bar
b$ and charmonium $c\bar c$ states in a constituent quark model where the
Cornell-like potential and spin-dependent interaction are employed, with all
model parameters predetermined by studying ground and first radial excited
states of S- and P-wave heavy quarkonium mesons. By comparing the theoretical
predictions for $J^{PC}=1^{--}$ quarkonium states with experimental data and
considering possible mixtures of $nS$ and $(n-1)D$ states, we provide tentative
assignments for all observed $J^{PC}=1^{--}$ heavy quarkonia. The work suggests
that the $\Upsilon$(10860) and $\Upsilon$(11020) are $b\bar b$ $5S-4D$ mixture
states, and the $\psi$(4360) and $\psi$(4415) are largely $4S$ and $3D$ $c\bar
c$ states respectively. The $\psi$(4260) may not be accommodated with the
conventional meson picture in the present work.
