Prepare a boiling-water canner with a rack; wash 4 pint jars, lids, and bands. Keep jars hot in the simmering canner water until ready to fill.
In a large bowl, mix the lemon juice with 4 cups cold water to make an acidulated soak; set aside.
Bring a large pot of water to a boil. Score a shallow X on the blossom end of each fruit. Blanch 30–60 seconds, transfer to the acidulated bowl, slip off skins, then halve and pit; keep halves submerged while you work.
In a wide nonreactive pot, combine the granulated sugar, apple cider vinegar (5% acidity), 2 cups water, cinnamon sticks, and whole cloves. Bring to a boil over medium-high heat, stirring to dissolve the sugar; boil gently 5 minutes.
Drain the peach halves from the acidulated water and add them to the boiling spiced syrup. Simmer until just tender but still holding shape, 5–8 minutes.
Working one jar at a time, pack hot halves cavity side down into hot jars, leaving 1/2 inch headspace. Ladle hot syrup and spices over to cover, maintaining 1/2 inch headspace. Remove air bubbles, adjust headspace, and wipe rims clean.
Apply lids and bands until fingertip-tight. Process in a boiling-water canner for 10 minutes (start timing once water returns to a full rolling boil).
Turn off heat and let jars rest in the water 5 minutes. Remove to a towel and cool 12–24 hours. Check seals, label, and store in a cool, dark place. Let cure at least 1 week before opening; refrigerate after opening.