Fairmagic´s Impossible-Bill-Twist

Fairmagic´s Impossible-Bill-Twist!
Fold a Signed Dollar Bill into an Impossible Souvenir!
You show the old gag where with two folding actions a dollar bill is turned on its head. You now have the spectator sign the bill and repeat the folding actions. However, now only the front of the bill is upside down, the back is not upside down. In other words, only one side of the bill turned. How is that possible? You give the dollar bill as souvenir to the spectator who can keep this impossible bill for the rest of his life in his wallet.

Remember that this is not some fake misprinted bill. I will teach you how to prepare a real genuine dollar bill for this effect.