Ask your butcher to cut pig into 3"–4" pieces (including feet), trim ears from head, and quarter head. Discard kidneys and upper portion of head with brain and eyes. Put pork into a very large bowl and set aside. Crush garlic and 1 tbsp. salt together with a mortar and pestle to a smooth paste. Stir in bitter orange juice and leaves from 3 sprigs of the oregano, then add to pork. Turn pork in marinade until well coated, cover, and refrigerate overnight.
The next day, chop 1 of the bunches of chives and put into a stockpot. Add peppers, culantro, cilantro, bay leaves, leeks, onions, allspice, and 6 quarts water, cover, and bring to a boil over high heat. Add pork and marinade and return liquid to a boil. Reduce heat to medium and simmer, partially covered, skimming foam as it rises to the surface, until pork is fork tender and broth has reduced by about a quarter, 2–3 hours. Transfer pork with a slotted spoon to a large bowl and set aside for frying. Strain broth into a large pot, discarding solids. Season to taste with salt and keep warm over low heat. To serve broth, divide the remaining chives and oregano between 6–8 sturdy glasses and ladle in some of the hot broth. Serve with lime wedges.
Heat lard in a large wide deep pot over medium-high heat until hot but not smoking (about 350º on a candy thermometer) Meanwhile, discard any solids still sticking to the pork, then pat pieces dry with paper towels. Fry pork in batches, stirring often with with a slotted metal spoon until crisp and deep golden brown, about 10 minutes per batch. Transfer pork with slotted spoon to a large platter and serve with garlic citrus sauce, if you like.