The spectrum of charmonium resonances contains a number of unanticipated states along with several conventional quark-model excitations. The hadrons of different quantum numbers $J^P$ appear in a fairly narrow energy band, where $J^P$ refers to the spin-parity of a hadron at rest. This poses a challenge for Lattice QCD studies of (coupled-channel) meson-meson scattering aimed at the determination of scattering amplitudes and resonance pole positions. A wealth of information for this purpose can be obtained from the lattice spectra in frames with nonzero total momentum. These are particularly dense since hadrons with different $J^P$ contribute to any given lattice irreducible representation. This is because $J^P$ is not a good quantum number in flight, and also because the continuum symmetry is reduced on the lattice. In this paper we address the assignment of the underlying continuum $J^P$ quantum numbers to charmonia in flight using a $N_f = 2 + 1$ CLS ensemble. As a first step, we apply the single-hadron approach, where only interpolating fields of quark-antiquark type are used. The approach follows techniques previously applied to the light meson spectrum by the Hadron Spectrum Collaboration. The resulting spectra of charmonia with assigned $J^P$ will provide valuable information for the parameterization of (resonant) amplitudes in future determinations of resonance properties with lattice QCD.
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