Based on the hypothesis that the X(3872) exotic hadron is a mixture of χc1(2P) and other states and that its prompt hadroproduction predominately proceeds via its χc1(2P) component, we calculate the prompt−X(3872) polarization at the CERN LHC through next-to-leading order in αs within the factorization formalism of nonrelativistic QCD, including both the color-singlet 3P[1]1 and color-octet 3S[8]1 c¯c Fock states. We also consider the polarization of the J/ψ 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/ψ 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 nonzero spin.