In a large pot, bring apple juice and cinnamon stick to a boil. Reduce heat; cover and simmer for 5 minutes. Stir in the pineapple juice, honey, lemon juice, lemon peel, and nutmeg and bring to a boil once more. Reduce heat; cover and simmer for 5 minutes. Discard cinnamon stick before serving.
While cider is simmering, use a toothpick to poke holes along the rind of each orange slice (spaced approximately 1/2-inch apart), and insert a clove into each hole.
Pour apple cider in a mug and serve hot with an orange slice garnish and an additional cinnamon stick, if you desire.
If desired, prepare cider on the stove and then transfer to a slow cooker set on low to keep warm while serving.