1. Rinse pork; pat dry. Cut garlic into 1/2-inch slices. Cut 1/2-inch slits all over meat; insert garlic in them. Place meat in a 1-gallon heavy plastic food bag.
2. In a blender, whirl orange juice, chilies with sauce, chopped oregano, and oil until smooth. Pour over pork, seal bag, and turn to coat. Set bag in a bowl. Chill at least 4 hours or up to 1 day, turning occasionally.
3. Prepare barbecue. If using charcoal briquets, mound and ignite 60 briquets on the firegrate of a barbecue with a lid. When briquets are dotted with gray ash, in 15 to 20 minutes, push equal amounts to opposite sides of firegrate. Add 5 more briquets now, and every 30 minutes of cooking, to each mound of coals.
If using a gas barbecue, turn heat to high and close lid for 10 minutes. Adjust burners for indirect cooking (no heat down center) and keep on high.
Set a drip pan on firegrate between coals or burners. Set barbecue grill in place.
4. Lift pork from marinade and lay on grill, not over heat. Cover barbecue; open vents for charcoal. During the first hour, baste meat with marinade occasionally. Cook pork until a thermometer reaches 155° in center of thickest part, 1 1/4 to 1 3/4 hours. Discard remaining marinade.
5. Transfer pork to a platter, keep warm, and let rest about 10 minutes. Garnish with oregano sprigs and orange wedges. Cut meat into thin slices, squeeze juice from orange wedges over slices, and add salt and pepper to taste.