  We discuss the double spin asymmetries in P-wave charmonium hadroproduction
with non-zero transverse momenta at fixed target energies, $\sqrt{s}\simeq40$
GeV, within the framework of the NRQCD factorization approach. The size of the
asymmetries and the projected statistical errors in a future option of HERA
with longitudinally polarized protons scattering off a polarized target
(HERA-$\vec N$) are calculated. Measurements of the $\chi_{c1}$ and $\chi_{c2}$
decays into dilepton plus photon should allow to distinguish between different
parametrizations for the polarized gluon distribution in the proton. At higher
energies ($\sqrt{s}=200$ GeV) the situation appears less favourable with the
presently envisaged integrated luminosities of the polarized RHIC collider.
