Many quantities of interest in communications, signal processing, artificial intelligence, and other areas can be expressed as the partition sum of some factor graph. Although the exact calculation of the partition sum is in many cases intractable, it can often be approximated rather well by the Bethe partition sum. In earlier work, we have shown that graph covers are a useful tool for expressing and analyzing the Bethe approximation. In this paper, we present a novel technique for analyzing double covers, a technique which ultimately leads to a deeper understanding of the Bethe approximation.