  Based on the hypothesis that the $X(3872)$ exotic hadron is a mixture of
$\chi_{c1}(2P)$ and other states and that its prompt hadroproduction
predominately proceeds via its $\chi_{c1}(2P)$ component, we calculate the
prompt-$X(3872)$ polarization at the CERN LHC through next-to-leading order in
$\alpha_s$ within the factorization formalism of nonrelativistic QCD, including
both the color-singlet $^3\!P_1^{[1]}$ and color-octet $^3\!S_1^{[8]}$ $c\bar
c$ Fock states. We also consider the polarization of the $J/\psi$ produced by
the subsequent $X(3872)$ decay. We predict that, under ATLAS, CMS, and LHCb
experimental conditions, the $X(3872)$ is largely longitudinally polarized,
while the $J/\psi$ is largely transversely polarized. We propose that the LHC
experiments perform such polarization measurements to pin down the nature of
the $X(3872)$ and other $X$, $Y$, $Z$ exotic states with non-zero spin.
