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Mix the Tex-Mex Rub ingredients in a small bowl. Sprinkle the chicken with the rub, inside and out, patting it onto the chicken to help it stick. Gently work your fingers under the skin on the breast, then rub some of the spices directly onto the breast meat. Fold the wingtips under the wings and truss the chicken. Skewer the chicken on the rotisserie spit, securing it with the spit forks. Let the chicken rest at room temperature until it is time to grill.
Set the grill up for indirect medium heat with the drip pan in the middle of the grill. On my Weber Summit, I preheat the grill with all burners on high, then turn off all but the outer burners, leaving them on high. Then I light the infrared rotisserie burner, and put the drip pan in the center of the grill, over the unlit burners.
Put the spit on the grill, start the motor spinning, and make sure the drip pan is centered beneath the chicken. Close the lid and cook with the lid closed as much as possible. I check the chicken after 30 minutes, and if it is browning well, I turn off the infrared rotisserie burner. Let the chicken cook until it reaches 160°F in the thickest part of the breast, about 1 hour and 20 minutes.
Remove the chicken from the rotisserie spit and remove the twine trussing the chicken. Be careful – the spit and forks are blazing hot. Let the chicken rest for 15 minutes, then carve and serve.