1. Grate 1 teaspoon rind and squeeze 5 tablespoons juice from limes. Place rind and 4 tablespoons juice in a medium bowl; place the remaining 1 tablespoon juice in a separate medium bowl. Grate 1 teaspoon grapefruit rind; section grapefruit. Grate 1 teaspoon tangerine rind; section tangerine. Grate 1 teaspoon orange rind; section orange. Add citrus sections to 1 tablespoon lime juice in bowl; set aside. Add rinds to lime rind mixture. Add 2 tablespoons oil, garlic, and chile to rind mixture; stir with a whisk.
2. Place chicken, breast side down, on a cutting board. Using poultry shears, cut along both sides of backbone, and open chicken like a book. Turn chicken breast side up; using the heel of your hand, press firmly against the breastbone until it cracks. Lift wing tips up and over back; tuck under chicken. Discard backbone and skin. Place chicken on a rimmed baking sheet. Spread citrus rind mixture evenly over chicken. Cover and refrigerate at least 4 hours.
3. Position oven rack in lower third of oven. Preheat broiler to high.
4. Sprinkle chicken with 1/2 teaspoon salt and 1/4 teaspoon black pepper. Broil chicken, breast side down, 20 minutes. Turn chicken over; broil an additional 20 minutes or until done, turning pan occasionally. Let stand 10 minutes.
5. Combine reserved citrus section mixture, remaining 2 tablespoons oil, remaining 1/4 teaspoon salt, remaining 1/8 teaspoon pepper, red onion, mint, and cilantro; toss to combine.