Rinse the hominy and place it in a large pot with water. Bring to a boil.

Thoroughly clean the pork head (this takes a bit of time). Add it to the pot with the hominy, along with the whole head of garlic and one whole peeled onion. Let cook for 1 hour.

After 1 hour, wash the pork leg and shank. Add them to the pot. Continue cooking for another 1 to 2 hours, depending on the heat level.

While the pozole is cooking, rinse, stem, and seed the ancho chiles. Boil them until soft, then blend and strain. Set aside.

Finely chop the remaining onion. Wash and slice the radishes. Thinly slice the cabbage, then wash and disinfect it. Cut the limes in half. Set all these aside for serving.

Watch the meat as it cooks—the leg and shank will cook faster than the head. If they are done, remove them from the pot so they don't fall apart while the head finishes cooking. Once everything is cooked, add salt, the blended chile, and sprinkle with dried oregano. Check the seasoning. Serve hot with the chopped vegetables and some tostadas.