Arrange rack in middle of oven; preheat to 400°F. Mix garlic, cumin, oil, paprika, pepper, oregano, 2 teaspoons salt, and finely grated zest from lemon in a medium bowl.
Cut lemon in half and squeeze out juice into spice mixture. Rub the cut lemons all over the turkey, and any remaining juice. Place lemons in cavity of turkey.
Place turkey in a roasting pan, or aluminum sheet pan with parchment underneath, breast side facing up. From both edges of cavity, loosen skin from breasts, being careful not to tear skin. Using your fingers, gently spread 3/4 of spice mixture under skin (reserve remaining spice mixture), then season turkey skin all over with remaining 1 teaspoon salt.
Roast turkey 10 minutes, then turn heat down to 375°F. Continue roasting, basting with remaining spice mixture and pan juices every 20 minutes, until thickest part of breast is pierced with instant-read thermometer and reads 155-165°F.
Transfer turkey to a cutting board and let rest 15 minutes, where its temperature will continue to rise to 170°F.
Purée cilantro, jalapeños, garlic, oil, lime juice, and salt in a blender until combined. Add mayonnaise and purée until well blended. Transfer to a small bowl, cover, and chill until ready to use.