Constructor's note: This was one of the first puzzles I ever made and the first puzzle I submitted to the NY Times (12 years ago and a much earlier and cruder version). A rejection letter turned me away from making puzzles for awhile, but I'm happy I got back into it. This is a revamped version of the puzzle with updated clues.
Fun fact: Vietnam and China share 10 of the zodiac calendar's 12 signs — the rat, tiger, dragon, snake, horse, goat, monkey, rooster, dog, and pig. But the Vietnamese honor the cat instead of the rabbit, and the water buffalo instead of the ox.