Beyond the animal it carries, a branch enters three fixed sets of relations that the almanac and the fate-calculation tradition use constantly. Wu belongs to the trio Yin, Wu and Xu, one of the four groups of three branches spaced four apart which are read as combining into Fire. It clashes with Zi, the branch six places away, and it forms one of the six harmonies with Wei. These relations are properties of position in the twelve, not of the animal.
The branches also do more work than the year cycle they are best known for. They divide the day into twelve double hours, so Wu names a period of two hours as well as a year and a month; they mark the twelve directions of the compass, Wu standing at south; and they carry the twelve stations against which the movement of Jupiter was tracked, which is one proposed origin for the twelve-year cycle itself.
The stem-and-branch day count is the longest continuously running day count known. Numbering days by ganzhi begins in the Shang inscriptions and has run without acknowledged interruption since, through every dynastic change and every calendar reform, because the cycle of sixty is independent of any astronomical quantity and therefore needs no correction. Chinese historical dates from the classical period onward can be checked against it.
Wu is the cardinal branch of the south and holds the noon double hour, standing opposite Zi at midnight. That opposition is the clearest of the six clashes, the pairings of branches six apart that the tradition reads as tension, and the Zi and Wu axis is used in ritual orientation and in siting as well as in the calendar.
Sources Chinese Mathematical Astrology: Reaching Out to the StarsHo Peng Yoke with The Chinese Sexagenary Cycle and the Ritual Origins of the CalendarAdam Smith