  We analyze the polarization observables of J/psi photoproduction at
next-to-leading order (NLO) within the factorization formalism of
nonrelativistic quantum chromodynamics (NRQCD). This is the first NLO study of
heavy-quarkonium polarization including the full relativistic corrections due
to the intermediate ^1S_0^{[8]}, ^3S_1^{[8]}, and ^3P_J^{[8]} color-octet (CO)
states in the worldwide endeavor to test NRQCD factorization at the quantum
level. We present theoretical predictions in the helicity, target, and
Collins-Soper frames of DESY HERA, evaluated using the CO long-distance matrix
elements previously extracted through a global fit to experimental data of
unpolarized J/psi production, and confront them with recent measurements by the
H1 and ZEUS Collaborations. We find the overall agreement to be satisfactory,
but the case for NRQCD to be not as strong as for the J/psi yield.
