We present new Herschel-SPIRE imaging spectroscopy (194-671 mu m) of the bright starburst galaxy M82. Covering the CO ladder from J = 4 arrow right 3 to J = 13 arrow right 12, spectra were obtained at multiple positions for a fully sampled ~3 x 3 arcmin map, including a longer exposure at the central position. We present measurements of super(12)CO, super(13)CO, CI, NII, HCN, and HCO+ in emission, along with OH+, H sub(2)O+, and HF in absorption and H sub(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 super(12)CO and super(13)CO, adding further evidence to the high-J lines tracing a much warmer (~500 K), less massive component than the low-/lines. The addition of super(13)CO (and CI) is new and indicates that CI may be tracing different gas than super(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 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.