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Special equipment: Meat grinder (or mixer with meat grinder attachment)
Place a large bowl, pusher and assembled meat grinder attachment fitted with a fine grinding plate in the freezer. In another large bowl, toss the pork shoulder, fatback, ancho chile powder, salt, garlic, chipotle powder, oregano, cumin, sugar and cayenne together until all the pork is completely coated with the spice mixture. Cover tightly with plastic wrap and place in the freezer to chill for up to 1 hour (you want the meat very cold but not frozen solid).
Mix together the vinegar and achiote paste in a small bowl with a fork until smooth and keep it in the refrigerator.
When ready to grind the meat, fill a large bowl with ice water and then nest the chilled bowl from the freezer into it. Working quickly, grind the pork into the chilled bowl with the machine at medium-high speed. Reserve any remaining spice mixture. After pork is ground, place it in the freezer until very cold, about 20 minutes (at 45 degrees F, the fat begins to melt and can affect the texture of the sausage).
Remove the ground pork from the freezer and add the chilled achiote mixture and any remaining spice mixture. Using clean hands (wear gloves to prevent staining) or the paddle attachment of a stand mixer, knead or mix the pork until a sticky mass forms, about 2 minutes by hand or 1 minute in a stand mixer at medium speed. Wrap the chorizo tightly in plastic wrap and refrigerate for 24 hours or up to 3 days before using (flavors will develop as the chorizo rests).
In a large nonstick skillet over medium-high heat, cook 1 pound (about a third) of the chorizo, breaking it into small pieces with the side of a wooden spoon. Stir occasionally until cooked through and browned, 5 to 7 minutes.
Serve the cooked chorizo on warm corn tortillas with shredded cabbage, pickled red onion, crumbled queso fresco, cilantro leaves and lime wedges. The chorizo can also be formed into patties or frozen for up to 3 months.