Oh I've thought about it. It won't be reversed. You are completely misunderstood here.
OK, one more time. And I'm really trying to be polite here. When you criss cross rotate a tire, you reverse it's direction. For example, when you move a tire from the left front to the right rear, you change it's rotational direction from counter clockwise to clockwise. (Just visualize this in your head).
You are accomplishing the exact same thing if you flip it. Now, knowing and understanding that, please explain what you think is happening differently if you flip it?
The OP's tires can be safely criss cross rotated because they are non directional. Knowing that, once they are, the criss cross rotation has induced the same directional difference as if the tire was flipped. Follow? Use your coin example if it helps you.