Like other starburst galaxies, M82 hosts compact, massive young star clusters that are interesting both in their own right and as benchmarks for population synthesis models. Can spectral synthesis models at resolutions around 1000 adequately reproduce the near-IR spectral features and the energy distribution of these clusters between 0.8 and 2.4 microns? How do the derived cluster properties compare with previous results from optical studies? We analyse the spectra of 5 massive clusters in M82, using data acquired with the spectrograph SpeX on the InfraRed Telescope Facility (NASA/IRTF) and a new population synthesis tool with a highly improved near-IR extension, based on a recent collection of empirical and theoretical spectra of red supergiant stars. We obtain excellent fits across the near-IR with models at quasi-solar metallicity and a solar neighbourhood extinction law. Spectroscopy breaks a strong degeneracy between age and extinction in the near-IR colours in the red supergiant-dominated phase of evolution. The estimated near-IR ages cluster between 9 and 30 Myr, i.e. the ages at which the molecular bands due to luminous red supergiants are strongest in the current models. They do not always agree with optical spectroscopic ages. Adding optical data sometimes leads to the rejection of the solar neighbourhood extinction law. This is not surprising considering small-scale structure around the clusters, but it has no significant effect on the near-IR based spectroscopic ages. [abridged]
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