Rat and Rooster compatibility
Rat (Zi, 子) and Rooster (You, 酉): a break (Po, 破).
These are fixed relations among the twelve branches of the Chinese calendar, set out in the classic San Ming Tong Hui (三命通会), written in 1578, quoted below. They describe the signs, not two people: no classic gives a method for matching two people by their signs, so there is no score here.
What the book says
Break (Po, 破)
Rat and Rooster are one of four pairs that break each other, in a list of forces the book gives near its end. It names the break and adds that clash and break can go either way.
In this force, Rabbit and Horse, Ox and Dragon, Rat and Rooster, Goat and Dog all break each other. Only Tiger, Monkey, Snake and Pig break by origin yet are in harmony, so they are not counted.
此煞，卯与午，丑与辰，子与酉，未与戌，皆相破。惟寅申巳亥原破，却三合，故不取
San Ming Tong Hui, volume 3: On the symbolic stars as a whole (总论诸神煞) · source text
So a clash or a break can be good or bad. They cannot be judged wholesale.
可见冲破有吉有凶，不可概论。
San Ming Tong Hui, volume 2: On the clash (论冲击) · source text
Their elements
The Rat is Water and the Rooster is Metal: Metal feeds Water.
How much it counts
A sign is one character of the eight in a BaZi chart, the year’s branch. The book reads these relations between the pillars of one chart, and tells the reader to look at the whole chart’s elements first:
Clash and break, three harmony and six harmony: where a chart has them, ask next how its five elements get on. Misfortune may give rise to blessing, and blessing to misfortune.
相冲相破，三合六合，命中有之，即求五行相得何如。或祸中生福，福中生祸
San Ming Tong Hui, volume 3: On the symbolic stars as a whole (总论诸神煞) · source text
Birth years, and 2027
- Rat: 1924, 1936, 1948, 1960, 1972, 1984, 1996, 2008, 2020. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Rat harms the year (Hai Tai Sui, 害太岁).
- Rooster: 1933, 1945, 1957, 1969, 1981, 1993, 2005, 2017. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Rooster has no clash with the year.
A sign’s year is usually counted from Chinese New Year, between late January and mid-February; BaZi counts it from the Start of Spring, about 4 February. Born in January or early February? You may be the sign before: the sign pages check your date.
Two real people?
Each person’s chart has eight characters, and the same relations run between all of them, not only the two signs. The two-chart page sets both charts side by side, branch by branch, and gives what each chart says about marriage on its own. No score there either.
Rat with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Dog · Pig.
Rooster with the other signs: Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig.