The cross over is to ensure that the tires wear evenly from side to side across the tread of the tire. Going front to back is for the same purpose. The front tires tend to wear on the outside of the tread while the rear tires tend to wear toward the center of the tread. The goal is to have them wear as flat as possible.
Just make sure the tires don't have direction tread pattern. If they do, your only option is to transfer them straight back. Most tires we use will be just fine to cross the fronts to the back and bring the backs straight forward.