Heavy hadron chiral perturbation theory (HHχPT) and XEFT are applied to the decays X(3872)→ψ(2S)γ and ψ(4040)→X(3872)γ under the assumption that the X(3872) is a molecular bound state of neutral charm mesons. In these decays the emitted photon energies are 181MeV and 165MeV, respectively, so HHχPT can be used to calculate the underlying D0D ̄0*+D̄0D0 *→ψ(2S)γ or ψ(4040)→(D0D ̄0*+D̄0D0*) γ transition. These amplitudes are matched onto XEFT to obtain decay rates. The decays receive contributions from both long-distance and short-distance processes. We study the polarization of the ψ(2S) in the decay X(3872)→ψ(2S)γ and the angular distribution of X(3872) in the decay ψ(4040)→X(3872)γ and find they can be used to differentiate between different decay mechanisms as well as discriminate between 2-+ and 1++ quantum number assignments of the X(3872). © 2011 American Physical Society.