Conservation of momentum is why the moons orbit increases as the earth slows down dispite taking more energy. If your speeding up the earths rotation you get energy from the lower orbit and angular momentum from the same place.
Though to be really accurate you need to talk about the center of mass of the earth moon system ect.
The moon can't move up or down without transferring momentum to the earth.
But the amount of momentum transferred and the amount of energy transferred don't match. The difference is expressed as heat in the tides on Earth.
So in one direction there is extra energy, in the other insufficient energy.