We have searched 101 Classical transneptunian objects for companions with the Hubble Space Telescope. Of these, at least 21 are binary. The heliocentric inclinations of the objects we observed range from 0.6-34 degrees. We find a very strong anticorrelation of binaries with inclination. Of the 58 targets that have inclinations of less than 5.5 degrees, 17 are binary, a binary fraction of 29 +7/-6%. All 17 are similar-brightness systems. On the contrary, only 4 of the 42 objects with inclinations greater than 5.5 degrees have satellites and only 1 of these is a similar-brightness binary. This striking dichotomy appears to agree with other indications that the low eccentricity, non-resonant Classical transneptunian objects include two overlapping populations with significantly different physical properties and dynamical histories.