The reason WHY they say you shouldn't change the direction of rotation on radial tires is this:

Radial tires have layers of webbed belts built into them (sometime made of steel) that are meant to only run in one direction. When you switch a tire to the opposite side of the car it changes the direction those belts are pulled which can cause them to break in some cases. That's when you might see some steel wires sticking out of the side of the tire, which indicates that a (steel) belt is broken and poked through the side wall.

The problem does not happen as often with tires that use other types of reinforcing (other than steel). Therefore some radials (not steel belted) can be switched from side to side. So it depends on the tire and how it is made.

Doc