Wash canning jars in hot soapy water, rinse thoroughly, and keep hot in a 200°F oven or in hot water until ready to fill, then place new lids in a small saucepan with water heated to 180°F without boiling.
Fill boiling water bath canner with enough water to cover jars by 1-2 inches and bring to a rolling boil, then prepare light syrup by boiling sugar and water together until sugar dissolves completely, keeping hot until needed.
Wash oranges thoroughly, peel completely removing all white pith, separate into segments, and remove all seeds and membrane pieces, working over a bowl to catch any juice.
Pack orange segments firmly into hot jars leaving exactly 1/2 inch headspace, then add 1 tablespoon bottled lemon juice to each pint jar or 1/2 tablespoon to each half-pint jar.
Cover orange segments with boiling water, reserved orange juice, or hot light syrup, maintaining 1/2 inch headspace and ensuring all fruit is completely covered with liquid.
Remove air bubbles using a plastic bubble removal tool, wipe jar rims clean with a damp cloth, place hot lids on jars, and apply screw bands fingertip tight.
Process filled jars in boiling water bath for 10 minutes for both pint and half-pint sizes, adding 5 minutes for altitudes 1,001-3,000 feet, 10 minutes for 3,001-6,000 feet, or 15 minutes for altitudes above 6,000 feet.
Turn off heat and let jars rest in canner for 5 minutes, then remove jars using jar lifter and place on a towel-lined counter to cool undisturbed for 12-24 hours before checking seals and storing.
