  We explore polarized heavy quarkonium production using the color evaporation
model at leading order. We present the polarized to total yield ratio as a
function of center of mass energy and rapidity in $p+p$ collisions. At energies
far above the $Q \overline Q$ production threshold, we find charmonium and
bottomonium production to be longitudinally polarized ($J_z=0$). The quarkonium
states are also longitudinally polarized at central rapidity, becoming
transversely polarized ($J_z=\pm1$) at the most forward rapidities.
