Place the oregano, peppercorns, chiles, bay leaf, canela, garlic, onion and orange in the center of a 1-foot square piece of cheesecloth. Gather up the edges and tie with kitchen twine to secure the aromatics inside the sachet.
Place the sachet, pork shoulder, pork belly, stock, brown sugar and 4 teaspoons salt in a large heavy pot. Bring to a boil over high heat, then cover, lower the heat to medium-low and simmer until the pork is tender and shreds easily, about 2 hours.
Remove and discard the sachet. Transfer the pork shoulder to a rimmed baking sheet.
Increase the heat to high and continue cooking the pork belly until only the fat remains and the pan juices are completely evaporated, about 15 minutes. Carefully transfer the pork shoulder back to the pot (it's ok if it breaks apart) and fry it in the rendered fat, turning occasionally and scraping the bottom of the pot, until browned on all sides and just beginning to crisp, 10 to 15 minutes; the shoulder will start to shred and that's ok. Most of the pieces should be bite-size with some smaller shredded and super-crisp bits in the mix. Transfer the carnitas with a slotted spoon to a paper-towel-lined baking sheet.
Serve the carnitas with tortillas, cilantro, onions, limes, salsa verde and pico de gallo.
Pice do Gallo:
Gently stir the tomatoes, chiles, garlic, onions, cilantro and lime juice in a medium bowl until completely combined. Taste and season with 1 1/4 teaspoons salt and more lime juice if needed. Let the pico sit uncovered to let the flavors meld, about 10 minutes.
Canela (Ceylon cinnamon) is the variety of cinnamon most commonly used in Mexico. The sticks are thinner and softer than the cassia cinnamon sticks used in the US and have a more delicate, sweet and floral flavor with less of the "heat" of the cassia.
For easier cleanup, transfer cooked belly and rendered fat to a large nonstick skillet and fry shoulder until browned and beginning to crisp.