  We study the polarization observables of J/psi hadroproduction at
next-to-leading order within the factorization formalism of nonrelativistic
quantum chromodynamics. We complete the present knowledge of the relativistic
corrections by also providing the contribution due to the intermediate
^3P_J^{[8]} color-octet states at this order, which turns out to be quite
significant. Exploiting the color-octet long-distance matrix elements
previously extracted through a global fit to experimental data of unpolarized
J/psi production, we provide theoretical predictions in the helicity and
Collins-Soper frames and compare them with data taken by CDF at Fermilab
Tevatron I and II and by ALICE at CERN LHC. The notorious CDF J/psi
polarization anomaly familiar from leading-order analyses persists at the
quantum level, while the situation looks promising for the LHC, which is bound
to bring final clarification.
