  We propose the measurement of $\chi_{cJ}$ polarizations at high energy hadron
colliders to study heavy quarkonium production mechanism. We find that the
color-singlet model in the $k_t$ factorization approach predicts very different
behavior for $\chi_{cJ}$ polarizations at the Fermilab Tevatron compared with
the NRQCD predictions in the collinear parton model. In the color-singlet $k_t$
factorization approach, for both $\chi_{c1}$ and $\chi_{c2}$ productions, the
helicity $h=0$ states dominate over other helicity states at large $p_T$. These
properties are very useful in distinguishing between the two production
mechanisms which are related to the interesting issue of $J\psi$ and $\psi'$
polarizations, and may provide a crucial test for the $k_t$ factorization
approach.
