  Prompt $\chi_c$ production at hadron colliders may provide a unique test for
the color-octet mechanism in nonrelativistic QCD. We present an analysis for
the polarization observables of $\chi_{c1}$ and $\chi_{c2}$ at next-to-leading
order in $\alpha_S$, and propose to measure them at the LHC, which is expected
to be important for testing the validity of NRQCD.
