  We study contributions from color-octet quarkonium formation mechanisms to
$J/\psi$ hadroproduction at low $p_\perp$. We include transitions of
color-octet $c\bar{c}$ states into ``direct'' $J/\psi$ and into $\chi_{1,2}$
which decay radiatively into a $J/\psi$. Together with earlier work, this
calculation constitutes a complete analysis of $p_\perp$-integrated $J/\psi$
production at leading twist. We find that the leading-twist contribution is not
sufficient to reproduce the observed production rates and polarization of the
$J/\psi$ and $\chi_{1,2}$. Hence there must exist other important quarkonium
production mechanisms at low $p_\perp$.
