  We calculate the ${\cal O}(\alpha_s^3)$ short-distance, QCD
collinear-factorized coefficient functions for all partonic channels that
include the production of a heavy quark pair at short distances. This provides
the first power correction to the collinear-factorized inclusive hadronic
production of heavy quarkonia at large transverse momentum, $p_T$, including
the full leading-order perturbative contributions to the production of heavy
quark pairs in all color and spin states employed in NRQCD treatments of this
process. We discuss the role of the first power correction in the production
rates and the polarizations of heavy quarkonia in high energy hadronic
collisions. The consistency of QCD collinear factorization and non-relativistic
QCD factorization applied to heavy quarkonium production is also discussed.
