Heat oil in a large skillet over medium heat; add diced pork, cumin, and salt and pepper to taste. Cook, stirring frequently, until evenly browned.
Transfer pork to a large pot. Stir in hominy and 3 cups water. Cook over medium heat for 1 hour.
While pork and hominy are cooking, place chiles in a small saucepan with 2 cups water. Boil for 45 minutes, covered. Drain, and remove stems. Place chiles in a blender with 1/4 cup water, onion, and garlic. Blend until smooth (you may still see seeds). Set aside.
Serve pork and hominy soup in bowls, garnished with lime juice, cilantro, cabbage. For spicy soup, stir one teaspoon of chile into each bowl of soup.
