Combine in a medium bowl 2 tablespoons of orange juice, 2 tablespoons of lime juice, oil, 2 tablespoons of onions, 1/2 teaspoon of salt, and pepper. Add the chicken, turning to coat. Cover and refrigerate for at least 1 hour or up to 4 hours.
In another medium bowl, combine the mango, 2 tablespoons reserved orange juice, 1 tablespoon reserved lime juice, 3 tablespoons reserved onion, 1/4 teaspoon reserved salt, and hot pepper sauce. Stir in the cilantro or basil.
Coat a grill rack with fat-free cooking spray. Preheat the grill to medium. Grill the chicken 4 inches from the heat, basting often with the marinade and turning once, until an instant-read thermometer registers 160 degrees F in the thickest part and the juices run clear, (3 to 4 minutes per side). Serve with the salsa.