Determining the spectrum of photons emitted by the quark-gluon plasma nonperturbatively remains an open computational challenge. In this paper, we calculate two moments of that spectrum at a tem-perature $T\approx 254\mathrm{MeV}$, employing lattice QCD with two flavors of $\mathcal{O}(a)$-improved Wilson fermions, without facing an inverse problem. Our central value for the difference of these two moments, which is sensitive to photon energies $\omega \gtrsim \pi T$, is lower than, but compatible with that obtained by integrating the leading-order weak-coupling photon spectrum. This study informs the “direct photon puzzle” in heavy-ion collision phenomenology, where it has proved difficult to simultaneously explain the yield and azimuthal anisotropy of photons not originating from final-state hadronic decays.