1Pour water into a large saucepan and bring to a boil. Remove from heat and toss in the orange spice tea bags. Cover and let steep 5 minutes. Remove tea bags and stir in sugar, apple cider, rum, and 2 of the cinnamon sticks. Heat just to steaming -- do not boil.
2Ladle hot cider into 6 mugs and drop 1/2 teaspoon butter into each. Garnish with a cinnamon "swizzle" stick.