Chopped avocado, fresh cilantro, and white onion; thinly sliced cabbage, jalapeños, and radishes; lime wedges.
New Mexico chiles are sold at Latin markets, specialty foods stores, and some supermarkets. Look for dried hominy at natural foods stores and ranchogordo.com.
Red Chile Purée.
Preheat oven to 400°. Rinse and dry chiles; place in a single layer on a rimmed baking sheet and roast, turning occasionally, until puffed, fragrant, and a slightly darker red, 5–6 minutes. Let cool.
Wearing gloves, use scissors to stem and halve chiles lengthwise. Discard seeds for less heat. Transfer to a medium saucepan and cover with 6 cups water. Add onion and garlic; season with salt. Bring to a simmer. Cook until chiles are soft, 25–30 minutes.
Drain chile mixture, reserving liquid. Purée mixture and 1 1/2 cups liquid in a blender until smooth, adding more liquid if needed for a sauce that can coat a spoon. Strain through a medium-mesh sieve into a medium bowl. DO AHEAD Purée can be made 5 days ahead. Cover; chill.
Season pork shoulder and ham hocks with 1 tablespoon salt. Rub garlic, chili powder, and cumin all over pork; set aside. DO AHEAD Pork can be marinated 1 day ahead. Cover and chill.
Drain hominy. Transfer to a large heavy pot; add 12 cups water. Stud each onion with 1 clove; add to pot with 2 tablespoons salt and bay leaves. Cover; bring to a boil. Reduce heat to medium-low and simmer, covered and stirring often, until hominy begins to soften, about 1 hour (hominy takes longer to cook than pork, so give it a jump start).
Add pork shoulder and ham hocks to hominy; pour in water to cover by 1 inch. Cook, uncovered, stirring occasionally and adding more water as needed to keep ingredients submerged, until hominy is tender and pork is very tender and falling apart, 2–2 1/2 hours. Remove onions, bay leaves, and hocks. Discard onions and let hocks cool slightly; pick meat from bones and discard any cartilage, skin, and fat. Return meat to pot and stir in 3/4 cup red chile purée. Season to taste with salt and more red chile purée, if desired.
Divide posole among bowls. Serve with remaining red chile purée and garnishes.