Using a vegetable peeler, remove two 4-inch long strips of orange peel from orange. Set aside. Halve the orange. Cut one orange half into 8 very thin slices (if necessary, remove any seeds); set the slices and the other orange half aside.
For the spice bag, place the strips of orange peel, one stick cinnamon, the the allspice, the cloves and the 1 piece star anise in the center of a double-thick, 6-inch-square piece of 100-percent cotton cheesecloth. Bring the corners of the cheesecloth together and tie securely with a clean string.
In a large saucepan combine cider, sugar, reserved orange half and the spice bag. Bring just to boiling; reduce heat. Cover and simmer for 20 minutes. Do not stir.
To serve, use a slotted spoon to remove the spice bag and orange half; discard. Stir vermouth into cider mixture. Ladle hot cider into mugs. Garnish with reserved orange slices, remaining cinnamon sticks and, if desired, whole star anise. Makes 8 servings.