  Transverse polarization of ${}^3 S_1$ charmonium states, produced directly in
$p\bar{p}$ collisions at asymptotically large transverse momentum p_t, has
emerged as the most prominent test of color octet contributions and spin
symmetry in quarkonium production. We present predictions for the polar angle
distribution at moderate values of p_t \sim 4 - 20 GeV, covered by the Tevatron
Run I data. We update the fits of NRQCD matrix elements and discuss their
theoretical uncertainties. With our best fit values, no transverse polarization
is expected at p_t \sim 5 GeV, but the angular distribution is predicted to
change dramatically as p_t increases to 20 GeV.
