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.