Chinese zodiac compatibility
Two signs can join, clash, punish, harm or break each other. These are fixed relations between the twelve branches of the Chinese calendar, and the classic San Ming Tong Hui (三命通会), written in 1578, gives the reason for each.
There are no scores here. No classic gives a method for matching two people by their signs. Each pair’s page gives what the book actually says.
Six harmonies (Liu He, 六合)
Each yang sign paired with a yin one: six pairs.
Three harmonies (San He, 三合)
Four groups of three signs, four apart, each making one element: Monkey, Rat and Dragon (Water); Pig, Rabbit and Goat (Wood); Tiger, Horse and Dog (Fire); Snake, Rooster and Ox (Metal). The book says a group needs all three.
Clashes (Chong, 冲)
Signs directly opposite each other, six apart.
Harms (Hai, 害)
Each befriends what the other clashes with.
Punishments (Xing, 刑)
Rat and Rabbit, the punishment of no propriety; Tiger, Snake and Monkey, of no gratitude; Ox, Dog and Goat, of trading on power; and Dragon, Horse, Rooster and Pig each punish their own kind.
Breaks (Po, 破)
The book lists four pairs. Folk lists add Tiger and Pig, and Snake and Monkey; the book says those two pairs are in harmony and not to count them.
What about the animals themselves?
Most zodiac compatibility is written from the animals’ personalities. San Ming Tong Hui’s section on the zodiac animals sets out the element reasoning, then asks a run of questions, and concludes that the real animals do not follow the elements at all:
Yet the Horse is the horse and the Rat the rat, the Rooster the rooster and the Rabbit the rabbit. If Water overcomes Fire, why does the rat not chase the horse? If Metal overcomes Wood, why does the rooster not peck the rabbit?
午，马也，子，鼠也，酉，鸡也，卯，兔也，水胜火，鼠何不逐马？金胜木，鸡何不啄兔？
Matched by how their elements overcome each other, the creatures of the twelve branches do not correspond at all. Wherever blooded creatures master one another, even to eating one another, it comes down to sharp or blunt teeth and strong or weak sinews.
十二辰之禽以气性相克则尤不相应。大凡含血之虫相服，至于相啖食者，以齿牙钝利、筋力优劣自相胜服也。
San Ming Tong Hui, volume 2: On the animals of the branches (论地支属相) · source text
So these pages give the relations between the branches, not the animals’ characters.
Every pair, by sign
Rat: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Ox: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Tiger: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Rabbit: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Dragon: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Snake: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Horse: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Goat: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Monkey: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Rooster: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Dog: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Pig: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig
Two real people?
A sign is one character of the eight in a BaZi chart. The two-chart page compares both charts in full, branch by branch.