1. In a blender or a food processor, whirl 2 cloves garlic, 2 tablespoons salad oil, 6 tablespoons vinegar, and achiote paste until smoothly puréed.
2. Rinse chicken, pat dry, and pull off and discard any lumps of fat. Rub both halves on all sides with half the achiote mixture. Set chicken pieces, skin up and side by side, in a 9- by 13-inch pan.
3. Bake in a 375° oven for 30 minutes. Remove from oven and quickly spread skin sides equally with remaining achiote mixture. Continue baking, occasionally basting with pan juices, until skin is richly browned (under paste), 30 to 45 minutes longer.
4. Meanwhile, peel and thinly slice onions; mince remaining 2 cloves garlic. In a 12-inch frying pan or a 5- to 6-quart pan over medium-high heat, combine onions, garlic, and remaining 1 tablespoon salad oil; stir often until vegetables are limp, 8 to 10 minutes. Add remaining 6 tablespoons vinegar and stir occasionally until liquid is evaporated and onions are lightly browned, 10 to 15 minutes longer. Add oregano, mint, and salt and pepper to taste. Spread onion mixture on a platter and keep warm.
5. Set chicken halves on onion mixture. Tear or cut meat from bones.