Leave 1- to 2-inch stems on the beets with the roots, if any. Scrub throroughly and cook in water to cover 30 to 45 minutes, or until tender. Drain, saving 1½ cups of the juice.
Plunge the beets at once into cold water. Peel. Discard the roots and the stems with the skins. Cut beets into ¼-inch slices.
Separate the onion into thin rings. Put a layer of beets in a shallow pan. Cover with some of the onion rings. Repeat the procedure unitl all peets and onion are used.
Warm the vinegar, beet juice and the cloves. Pour mixture over the beets. Chill overnight to blend the flavors.