Place the cinnamon sticks, cloves, and allspice berries onto the center of a 8 inch square piece of cheesecloth. Gather together the edges of the cheesecloth, and tie with kitchen twine to secure. Place spice sachet into a large saucepan and pour in the vinegar. Bring to a simmer over medium heat; cook for 5 minutes. Stir blueberries into the vinegar; cook until heated through, about 5 minutes. As the blueberries heat, gently shake the pot. Do not stir to avoid breaking the berries. Remove from heat, cover, and let stand at room temperature for 8 to 12 hours.
Pour berries and liquid into a colander set over a bowl. Remove spice sachet. Transfer berries to hot, sterilized canning jars; reserve the liquid. Return liquid to saucepan and place over high heat. Stir in the white and brown sugars; bring to a boil. Boil until thickened, about 4 minutes.
Ladle hot syrup over berries, leaving 1/2 inch headspace. Wipe the rims of the jars with a moist paper towel to remove any food residue. Top with sterilized lids; screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil over high heat, then carefully lower the jars into the pot. Leave a 2 inch space between the jars. Pour in more boiling water if necessary until the water level is at least 1 inch above the tops of the jars. Bring the water to a full boil, cover the pot, and process for 15 minutes.
