Cook pork: Put all ingredients for pork in a large heavy pot and add enough water to cover pork. Bring to a boil, skimming foam periodically, then reduce to a simmer. Cover and simmer until a paring knife slides in easily, 1 1/2 to 2 hours. Remove from heat and let cool. Lift out meat and onions (reserve broth for tamale dough if you like, and the onions for soup) and tear meat into bite-size shreds, discarding bones and excess fat. Rinse out pot and reserve.
Make salsa: Snip chiles open with scissors. Remove stem, seeds, and any white membranes. Add chiles, onion, and garlic to a pot with 3 1/2 cups water. Bring to a boil and remove from heat. Stir in cloves, cumin, and 1/2 tsp. salt. In 2 batches, whirl in a blender until smooth.
In same pot you used for pork, heat oil over medium-high heat. Pour in salsa and bring to a boil, then reduce to a simmer and cook 15 minutes. Add shredded pork, remaining 1/2 tsp. salt, and the pepper and cook until salsa is thick and clingy and flavors are incorporated, about 15 minutes. Season to taste with salt and pepper.
Make ahead: Up to 2 days, chilled.