  We discuss exclusive elastic double diffractive axial-vector $\chi_c(1^+)$
and tensor $\chi_c(2^+)$ mesons production for different meson polarisations in
proton-(anti)proton collisions at the Tevatron energy. The amplitude for the
process is derived within the $k_t$-factorisation approach using unintegrated
gluon distributions (UGDFs). Differential cross sections for different $\chi_c$
polarisations are calculated. Angular distributions of $J/\psi$ meson from the
radiative $\chi_c(1^+,2^+)$ decays are derived. Prospects for experimental
selection of different spin states of $\chi_c$ mesons are discussed.
