Margaret, This is the way my mother-in-law used to pickle beets, and she always had them for me because she knew I liked them. She canned hers, too. When she had finished with the beets, then she put hard boiled eggs in the pickling liquid, and pickled them until they were a pretty rosy color.

Pickled Beets

Beets--8 medium raw or whole canned 1-2 cans
1 cup apple cider vinegar (any vinegar will do)
2 cups beet cooking water (or juice from canned beets + water to 2 cups
1/2 cup sugar (or more to your taste, balancing sweet & sour)
1 medium onion
1 bay leaf
3-6 whole cloves or 1/4 -1/2 tsp. powdered (use only new strong spices)
salt to taste cinnamon (bark or powder) if desired, to taste.

Parboil beets (cut off root & leaves) & slip off skins, or just peel raw.
Cook beets whole if small, sliced if large, in salted water until fork
tender. Transfer beets to glass/china container with lid, size large
enough to completely cover beets & allow room for 2-4 more cups displacement.
In cooking water pour vinegar, sugar, spices, seasonings. Taste for
comfortable acidity & add either more water or more vinegar; taste for
balance of sweet & sour you like & add sugar if wanted, or readjust
water/vinegar to make comfortably sour. Peel & thinly slice raw onion into
beets & juice. Cover & let sit out until room temperature, then
add peeled harboiled eggs. Refrigerate for 24 hours. Improves each day.
(my grandmother used to have this dish of beets & a bowl of homemade
applesauce always on hand, summer & winter..low calorie & satisfying)