  Starting from the Feynman diagram representation of multiple scattering we
consider the polarized $\chi_c$(1P)-charmonia production in antiproton-nucleus
reactions close to the threshold ($p_{\rm lab}=5-7$ 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 $\chi_{cJ} N \to \chi_{cJ^\prime}
N$, $J,J^\prime=0,1,2$. The elementary amplitudes of the latter processes are
evaluated by expanding the physical $\chi_c$-states in the Clebsch-Gordan
series of the $c \bar c$ states with fixed values of internal orbital angular
momentum ($L_z$) and spin projections on the $\chi_c$ momentum axis. The total
interaction cross sections of these $c \bar c$ 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 $\chi_c$-nucleon nondiagonal scattering amplitudes. We show that the
$\chi_{c0} N \to \chi_{c2} N$ transitions significantly influence the
$\chi_{c2}$-production with helicity zero at small transverse momenta. This can
serve as a signal in future experimental tests of the quark structure of
$\chi_c$-states by the PANDA collaboration at FAIR.
