  We show that when the factorized cross section for heavy quarkonium
production includes next-to-leading power (NLP) contributions associated with
the production of the heavy quark pair at short distances, it naturally
reproduces all high $p_T$ results calculated in non-relativistic QCD (NRQCD)
factorization. This extended formalism requires fragmentation functions for
heavy quark pairs, as well as for light partons. When these fragmentation
functions are themselves calculated using NRQCD, we find that two of the four
leading NRQCD production channels, ${^3\hspace{-0.6mm}S_{1}^{[1]}}$ and
${^1\hspace{-0.6mm}S_{0}^{[8]}}$, are dominated by the NLP contributions for a
very wide $p_T$ range. The large next-to-leading order corrections of NRQCD are
absorbed into the leading order of the first power correction. The impact of
this finding on the heavy quarkonium production and its polarization is
discussed.
