Bring the vinegar, 3/4 cup of the pickling liquid, the onions, 4 teaspoons salt, the mustard seeds, coriander seeds and sugar to a boil in a medium saucepan over medium heat, stirring to dissolve the salt and sugar. Reduce the heat, cover and let the brine simmer for 5 minutes.
Meanwhile, pierce each egg all the way through about 6 times with a cake tester or toothpick. Put 3 eggs and 1/4 of the beets in each of 2 glass quart jars with lids. Ladle in enough brine to cover. Add 3 more eggs to each jar, divide the remaining beets between them and ladle in more brine to cover, leaving about 1/4 inch of space at the top. Close the jars, let cool to room temperature, then refrigerate for at least 24 hours before serving.
Serve the eggs and beets halved or quartered, with some of the onions and pickling liquid spooned on top. (The eggs will keep, covered by liquid and refrigerated, up to 5 days.)