We present the first X-ray observations of the Taffy galaxies (UGC 12914/5) with the Chandra observatory, and detect soft X-ray emission in the region of the gas-rich, radio-continuum-emitting Taffy bridge. The results are compared to Herschel observations of dust and diffuse [CII] line-emitting gas. The diffuse component of the Taffy bridge has an X-ray luminosity of L(X) (0.5-8keV) =5.4 x 10^39 erg s^-1, which accounts for 19% of the luminosity of the sum for the two galaxies. The total mass in hot gas is (0.8--1.3) x 10^8 M_sun, which is approximately 1% of the total (HI~+~H2) gas mass in the bridge, and ~11% of the mass of warm molecular hydrogen discovered by Spitzer. The soft X-ray and dense CO-emitting gas in the bridge have offset distributions, with the X-rays peaking along the north-western side of the bridge in the region where stronger far-IR dust and diffuse [CII] gas is observed by Herschel. We detect nine Ultra Luminous X-ray sources (ULXs) in the system, the brightest of which is found in the bridge, associated with an extragalactic HII region. We suggest that the X-ray--emitting gas has been shocked--heated to high temperatures and "splashed" into the bridge by the collision. The large amount of gas stripped from the galaxies into the bridge and its very long gas depletion timescale (>10 Gyr) may explain why this system, unlike most major mergers, is not a powerful IR emitter. ; Comment: Accepted for Publication in ApJ, Sept 15 2015