We report the detection of strong , resolved emission from warm { H _ { 2 } } in the Taffy galaxies and bridge . Relative to the continuum and faint PAH emission , the { H _ { 2 } } emission is the strongest in the connecting bridge , approaching L ( H _ { 2 } ) / L ( PAH8 \micron ) = 0.1 between the two galaxies , where the purely rotational lines of { H _ { 2 } } dominate the mid-infrared spectrum in a way very reminiscent of the group-wide shock in the interacting group Stephan ’ s Quintet . The surface brightness in the 0–0 S ( 0 ) and S ( 1 ) { H _ { 2 } } lines in the bridge is more than twice that observed at the center of the Stephan ’ s Quintet shock . We observe a warm { H _ { 2 } } mass of 4.2 \times 10 ^ { 8 } M _ { \odot } 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 \sim 5 \times 10 ^ { 6 } M _ { \odot } kpc ^ { -2 } and typical excitation temperatures of 150–175 K . { H _ { 2 } } 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 { H _ { 2 } } is short ( \sim 5000 yr ) , shocks must be permeating the molecular gas in bridge region in order to continue heating the { H _ { 2 } } .