We present a compilation of published high-precision velocities for 107 isolated binary galaxies, which we use to obtain the distribution function of their velocity differences. The distribution shows a peak at the zero difference as expected, but it also shows a preference for values near 72 km s-1. The distribution function declines smoothly beyond approximately 72 km s-1, with no significant peaks at multiples of 72 km s-1 as claimed by Tifft. However, any peak in the velocity-difference distribution away from zero violates simple geometrical constraints for a complete sample of bound pairs. If neither statistical fluctuations nor "new physics" are the cause, one must consider whether selection procedures on binary galaxy samples may be responsible for the anomalous distribution. We point out that selection criteria defined on the basis of too narrow a projected separation on the sky can produce a nonmonotonic distribution if the orbits are eccentric. Such orbits can produce a strong secondary peak only if the level of incompleteness in binary-galaxy samples is quite high, suggesting that present statistical estimates of the masses of binary galaxies should be reevaluated. The two peaks in the distribution function further suggest that two populations of orbits may be present: dynamical friction in pairs that have already had a close encounter could lead to a peak at small DELTA-epsilon's in addition to or partially masking a peak at finite DELTA-upsilon in precollision pairs.