Remove and discard the neck, giblets, pop-up timer, and any excess fat from the turkey.
Prepare the grill for indirect cooking over medium heat (350° to 450°F).
Pierce each orange several times with a knife. Insert six whole cloves into each orange. Put the clove-studded oranges, garlic, and sage into the turkey cavity. Truss the turkey with butcher’s twine, tying the wings close to the body and securing the legs. Pin the skin flap to the body with two or three small trussing needles. Position one set of fork prongs on the far end of the spit and slide the spit into the neck end through the turkey (skewer the oranges) and out the other side, just underneath the drumsticks. Slide the other set of fork prongs on the spit and drive the prongs into the back of the turkey. Make sure the turkey is centered on the spit before tightening the fork prongs into place. Rub the entire exterior of the turkey with the rub, pressing it into the skin.
Set the spit in place, turn on the rotisserie motor, and let the turkey rotate over indirect medium heat, with the lid closed, until the internal temperature reaches 170º to 175ºF in the thickest part of the thigh (not touching the bone), 2¼ to 2½ hours. Note: If you do not have a rotisserie, grill the turkey over indirect medium heat for 2¼ to 2½ hours.
When the turkey is fully cooked, turn off the rotisserie motor and, wearing insulated barbecue mitts, carefully transfer the spit from the grill to a cutting board. Loosely cover with aluminum foil and let rest for at least 20 minutes (the internal temperature will rise a few degrees during this time). Remove the spit and discard the oranges, garlic, and sage. Carve the turkey and serve warm.