Set up the grill for rotisserie cooking and preheat to high.
Remove any lumps of fat in chicken cavity; wash inside and out, and blot dry. Rub chicken all over with half the cut head of garlic, then with half the cut lemon.
Mix ingredients for rub together. Sprinkle some rub into chicken and place remaining garlic and lemon halves inside. Truss chicken, brush outside with oil, and rub with remaining spice mix. Thread chicken onto spit and secure with prongs.
Cook chicken until skin is crisp and a deep golden brown and meat is cooked through, 1 1/4 to 1 1/2 hours. When cooked, internal temperature in thigh will be 175 degrees. Transfer chicken to a platter, let rest for 5 minutes and untruss. Carve or cut in half, and serve.