[en] We present new Herschel-SPIRE imaging spectroscopy (194-671 μm) of the bright starburst galaxy M82. Covering the CO ladder from J = 4 → 3 to J = 13 → 12, spectra were obtained at multiple positions for a fully sampled ∼3 × 3 arcmin map, including a longer exposure at the central position. We present measurements of ¹²CO, ¹³CO, [C I], [N II], HCN, and HCO⁺ in emission, along with OH⁺, H₂O⁺, and HF in absorption and H₂O in both emission and absorption, with discussion. We use a radiative transfer code and Bayesian likelihood analysis to model the temperature, density, column density, and filling factor of multiple components of molecular gas traced by ¹²CO and ¹³CO, adding further evidence to the high-J lines tracing a much warmer (∼500 K), less massive component than the low-J lines. The addition of ¹³CO (and [C I]) is new and indicates that [C I] may be tracing different gas than ¹²CO. No temperature/density gradients can be inferred from the map, indicating that the single-pointing spectrum is descriptive of the bulk properties of the galaxy. At such a high temperature, cooling is dominated by molecular hydrogen. Photon-dominated region (PDR) models require higher densities than those indicated by our Bayesian likelihood analysis in order to explain the high-J CO line ratios, though cosmic-ray-enhanced PDR models can do a better job reproducing the emission at lower densities. Shocks and turbulent heating are likely required to explain the bright high-J emission.