Peel garlic cloves and reserve 2 for chile sauce. Slice remaining garlic. In a 7 to 8 quart heavy kettle bring water and broth just to a boil with sliced garlic and pork. Skim surface and add oregano. Gently simmer pork, uncovered, until tender, about 1 1/2 hours.
While pork is simmering, wearing protective gloves, discard the stems from chiles and in a bowl combine chiles with boiling water. Soak chiles turning them occasionally for 30 minutes. Cut onion into large pieces. In a blender puree chiles, soaking liquid, onion, 2 teaspoons salt, reserved garlic until smooth.
Transfer pork with tongs to a cutting board and reserve broth mixture. Shred pork using 2 forks and discard bones. Rinse and drain hominy. Return pork to broth mixture and add chile sauce, hominy, and remaining teaspoon salt. Simmer pozole 30 minnutes.
Serve with corn chips, diced avocado, thinly sliced lettuce, lime wedges, dried oregano, dried hot red pepper flakes, any hot peppers you like.