Combine 1 cup beet juice, sugar, vinegar, cloves, salt, pepper, and bay leaves in a saucepan; bring to a boil. Reduce heat and simmer until sugar is dissolved, about 5 minutes.
Place beets and eggs into a large bowl. Pour beet juice mixture over beets and eggs. Refrigerate for 3 days. Drain and discard pickling liquid.
Cut each egg in half lengthwise and scoop yolks into a bowl. Mix mayonnaise, relish, and mustard into egg yolks until smooth. Spoon egg yolk filling back into each egg white. Serve beets and eggs on a platter.