  Gluon fragmentation represents the dominant source of high energy prompt
quarkonia at hadron colliders. Fragmentation approximations break down,
however, when a quarkonium's transverse momentum becomes comparable to its
mass. In this paper, we identify a large class of color-octet diagrams that
mediate quarkonia production at all energies and reduce to the dominant set of
gluon fragmentation graphs in the high $p_\perp$ limit. They contribute to
quarkonia differential cross sections at the same order as color-singlet
diagrams and bring theoretical predictions for Upsilon and Psi production at
the Tevatron into agreement with experimental measurements. Using recent CDF
data, we extract numerical values for bottomonia and charmonia color-octet
matrix elements which are consistent with NRQCD scaling rules. We also find
that quarkonia generated via the color-octet mechanism are strongly polarized
at low as well as high energies. Spin alignment measurements can thus test the
color-octet quarkonia production picture.
