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 x 3 arcminute map, including a longer exposure at the central position. We present measurements of 12 CO, 13 CO, [C I], [N ii], HCN, and HCO + in emission, along with OH + , H 2 O + and HF in absorption and H 2 O 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 12 CO and 13 CO, 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 13 CO (and [C I]) is new and indicates that [C I] may be tracing different gas than 12 CO. 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.