Cut several layers of cheesecloth into 8-inch squares. Cut 3 (1-inch) strips of peel each from orange and lemon; place in cheesecloth. Add cinnamon sticks and cloves. Wrap and secure with string.
Bring 4 cups water to a boil in a saucepan. Add spice packet and brown sugar, reduce heat, and simmer 10 minutes. Remove from heat, and add tea bags. Steep 5 minutes. Remove spice packet and tea bags. Stir in cider. Juice orange and lemon, and pour into cider mixture. Heat over medium heat until hot but not boiling.