Dragon and Dog compatibility
Dragon (Chen, 辰) and Dog (Xu, 戌): a clash (Chong, 冲).
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
Clash (Chong, 冲)
Dragon and Dog sit directly opposite each other on the circle of twelve, six signs apart. The book calls clashes broadly ill, and then names the ones that are not.
Among the branches the seventh along is the clash, just as among the stems the seventh along is the killing force.
地支取七位为冲，犹天干取七位为煞之义。
A clash is the twelve branches at war, striking at each other, and broadly it is ill. Yet some clashes bring the greatest blessing: those where the two sides also feed each other.
相冲者，十二支战击之神，大概为凶。然有为福之甚者，乃冲处相生。
Where the four stores, Dog, Ox, Dragon and Goat, hold the ten stems’ wealth, office, seal and the like, they especially welcome being clashed open.
若戌丑辰未四库所藏为十干财官、印绶等物，尤喜冲激
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 Dragon is Earth and the Dog is Earth: the same element.
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
- Dragon: 1928, 1940, 1952, 1964, 1976, 1988, 2000, 2012, 2024. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Dragon has no clash with the year.
- Dog: 1934, 1946, 1958, 1970, 1982, 1994, 2006, 2018. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Dog is punished by the year (Xing Tai Sui, 刑太岁) and breaks with the year (Po Tai Sui, 破太岁).
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.
Dragon with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Pig.
Dog with the other signs: Rat · Ox · Tiger · Rabbit · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig.