Preheat oven to 325 degrees F. Remove giblets from the turkey's cavity and save for another use. Rinse the turkey and use paper towels to dry inside the cavity. Rub the salt and 1 tablespoon chili powder in the cavity, then stuff with the oranges and garlic. Tuck the wings under the back and tie the legs. Rub with oil and remaining tablespoon chili powder.
Place on a rack in a shallow roasting pan. Roast on the lowest rack 3 to 3 3/4 hours or until an instant-read thermometer registers 180 degrees F when placed in thigh. Remove roasting pan from oven. Carefully lift turkey from roasting pan allowing any collected juice from the cavity to flow into the pan. Place the turkey on the platter and keep tented with foil to keep warm. Carve and serve warm, or use to make fajitas or Cuban sandwiches.