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 in its rest frame. 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 Coordinated Lattice Simulations ensemble. As a first step, we apply the singlehadron 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 JP will provide valuable information for the parametrization of (resonant) amplitudes in future determinations of resonance properties with lattice QCD.