Cut orange into 1/4-inch-thick rounds; cut rounds into 1/4-inch cubes. Place in heavy medium saucepan; add 3 cups water. Bring to boil. Reduce heat and simmer 10 minutes. Drain, rinse, and drain well. Boil again with 3 cups water; rinse and drain.
Return orange cubes to same pan. Add 3/4 cup water, 1/2 cup sugar, and 1 cinnamon stick. Bring to boil, stirring until sugar dissolves. Boil gently until all liquid evaporates, about 30 minutes.
Bring brown sugar, rum, vinegar, allspice, 1/4 cup water, and 1 cinnamon stick to boil in saucepan. Mix in cranberries. Cook until thick syrup forms, stirring occasionally, about 10 minutes. Mix in orange; cook 1 minute. Transfer chutney to small bowl; cool. (Can be made 2 days ahead. Cover; chill.)
