  We analyze the inclusive prompt production of $\psi^\prime$, $\chi_c$,
$J/\psi$ and $\eta_c$ mesons at the LHC using the $k_T$-factorization approach.
Our consideration is based on the off-shell production amplitudes for hard
partonic subprocesses, nonrelativistic QCD formalism for the formation of bound
states and transverse momentum dependent (or unintegrated) gluon densities in a
proton derived from Catani-Ciafaloni-Fiorani-Marchesini equation. The
nonperturbative color octet transitions are treated in terms of the multipole
radiation theory. We extract the corresponding long-distance matrix elements
from a combined fit to transverse momentum distributions measured at various
LHC experiments. We make predictions for the polarization parameters
$\lambda_\theta$, $\lambda_\phi$, $\lambda_{\theta \phi}$ and the
frame-independent parameter $\lambda^*$ and compare them to the available
$\psi^\prime$ and $J/\psi$ data. Finally, we present a universal set of
parameters that provides a reasonable simultaneous description for the whole
body of the LHC data (on the $p_T$ distributions, relative production rates and
polarization observables) for the whole charmonium family.
