A joint determination of the reactor antineutrino spectra resulting from the fission of U235 and Pu239 has been carried out by the Daya Bay and PROSPECT Collaborations. This Letter reports the level of consistency of U235 spectrum measurements from the two experiments and presents new results from a joint analysis of both data sets. The measurements are found to be consistent. The combined analysis reduces the degeneracy between the dominant U235 and Pu239 isotopes and improves the uncertainty of the U235 spectral shape to about 3%. The U235 and Pu239 antineutrino energy spectra are unfolded from the jointly deconvolved reactor spectra using the Wiener-SVD unfolding method, providing a data-based reference for other reactor antineutrino experiments and other applications. This is the first measurement of the U235 and Pu239 spectra based on the combination of experiments at low- and highly enriched uranium reactors.