We report the detection of strong, resolved emission from warm H2 in the Taffy galaxies and bridge. Relative to the continuum and faint PAH emission, the H2 emission is the strongest in the connecting bridge, approaching L(H2)/L(PAH8{mu}m) = 0.1 between the two galaxies, where the purely rotational lines of H2 dominate the mid-infrared spectrum in a way very reminiscent of the group-wide shock in the interacting group Stephans Quintet. The surface brightness in the 0-0 S(0) and S(1) H2 lines in the bridge is more than twice that observed at the center of the Stephans Quintet shock. We observe a warm H2 mass of 4.2 times 108 Modot in the bridge, but taking into account the unobserved bridge area, the total warm mass is likely to be twice this value. We use excitation diagrams to characterize the warm molecular gas, finding an average surface mass of 5 times 106 Modot kpc-2 and typical excitation temperatures of 150-175 K. H2 emission is also seen in the galaxy disks, although there the emission is more consistent with normal star forming galaxies. We investigate several possible heating mechanisms for the bridge gas, but favor the conversion of kinetic energy from the head-on collision via turbulence and shocks as the main heating source. Since the cooling time for the warm H2 is short (5000 yr), shocks must be permeating the molecular gas in bridge region in order to continue heating the H2.
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