Hypothetically, suppose there is a situation where the Earth's moon gets neatly sliced into two equal hemispheres, and the force responsible for this slicing also creates a distance between the two halves of say, half the (average) distance from the Earth to the Moon, as it originally stood.
Do the two halves revolve around each other? Do they rotate individually? Do they each still revolve around the Earth?
EDIT: The angular momentum of each half after the division is the same as the angular momentum of the moon before the division occurred.