Starting from the Feynman diagram representation of multiple scattering we consider the polarized χ c (1P)-charmonia production in antiproton-nucleus reactions close to the threshold (p lab = 5 − 7 GeV/c). The rescattering and absorption of the incoming antiproton and outgoing charmonium on nucleons are taken into account, including the possibility of the elastic and nondiagonal (flavor-conserving) scattering χ cJ N → χ cJ ′ N , J, J ′ = 0, 1, 2. The elementary amplitudes of the latter processes are evaluated by expanding the physical χ c -states in the Clebsch-Gordan series of the cc states with fixed values of internal orbital angular momentum (L z ) and spin projections on the χ c momentum axis. The total interaction cross sections of these cc states with nucleons have been calculated in previous works using the QCD factorization theorem and the nonrelativistic quarkonium model and turned out to be strongly L z -dependent due to the transverse size difference. This directly leads to finite values of the χ c -nucleon nondiagonal scattering amplitudes.We show that the χ c0 N → χ c2 N transitions significantly influence the χ c2 -production with helicity zero at small transverse momenta. This can serve as a signal in future experimental tests of the quark structure of χ c -states by the PANDA collaboration at FAIR.