We present further observations of molecular gas in head-on collisions of spiral galaxies, this time of the CO(J=1-0) and CO(J=2-1) lines in the UGC 813 -- UGC 816 system. UGC 813/6 are only the second known example of head-on spiral-spiral collisions, the first example being the UGC 12914/5 pair. Strong CO emission is present in the bridge between UGC 813 and 816, unassociated with stellar emission, just as in UGC 12914/5. The CO emission from the UGC 813/6 bridge, not counting the emission from the galaxies themselves, is at least that of the entire Milky Way. Collisions of gas-rich spirals are really collisions between the interstellar media (ISMs) of the galaxies. We show that collisions between molecular clouds bring H2 into the bridge region. Although the dense clouds are ionized by the collisions, they cool and recombine very quickly and become molecular again even before the galactic disks separate. Because the clouds acquire an intermediate velocity post-collision, they are left in the bridge between the separating galaxies. The star formation efficiency appears low in the molecular clouds in the bridges. We speculate that the pre-stellar cores in the molecular clouds may expand during the cloud collisions, thus retarding future star formation. Because the ISM-ISM collisions discussed here require a very small impact parameter, they are rare among field spirals. In clusters, however, these collisions should be an important means of ejecting enriched gas from the inner parts of spirals.