Using a vegetable peeler, remove a 1-inch-wide strip of orange peel from orange. Remove and discard the remaining peel. Halve the orange.
For the spice bag, place the orange peel, stick cinnamon, whole allspice, and whole cloves 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, orange halves, and the spice bag. Bring to boiling; reduce heat. Cover and simmer for 1 hour. Stir in vanilla.
To serve, dip rims of heatproof glasses or mugs in water, then in a mixture of the sugar and nutmeg. Use a slotted spoon to remove the spice bag and orange halves. Ladle hot cider into the glasses or mugs. Makes 4 (8-ounce) servings.
If you like, use a whole vanilla bean, halved lengthwise, in place of vanilla. Add to mixture with spice bag. Remove it before serving.