The Flipped Mess is a variation
of Mills Mess in which the ball at the front of the pattern, the one that is
normally both thrown and caught over-the-arm, is instead thrown and
caught under-the-arm. This leads to an awkward Mills Mess pattern
in which the third ball, normally thrown under-the-arm, is instead thrown
over-the-arm, meaning that the throw positions of the first and third ball
have been swapped, hence the name Flipped Mess.
To learn the Flipped Mess, start with two balls in your dominant hand and one ball in your non-dominant. Cross your dominant hand over your non-dominant hand (this is the same configuration you would start in for the regular Mills Mess), and then make a throw from your dominant hand back toward the dominant side your body. You will then make a throw from your non-dominant hand (still crossed underneath your dominant hand) from the center of your body toward the dominant side of your body, clearing space for the hand to catch the first ball. Finally, you are going to make a throw from your dominant hand toward the center of your body, and then catch the second ball with your now empty dominant hand. The third ball will be caught by your non-dominant hand, which should uncross before catching.
As shown, practice this on both sides. Notice again that when catching the final throw your non-dominant hand uncrosses. To connect both sides of the pattern together, you are going to make an under-the-arm throw from your non-dominant hand before it uncrosses to catch the other ball. The hand will then recross over your dominant hand. This position will set you up to repeat the process on the opposite side of your body, with your dominant hand making an under-the-arm throw and catching the ball thrown from your non-dominant hand. Your non-dominant hand will then make a final throw toward the non-dominant side of your body, catching the under-the-arm throw made by your dominant hand.
You have just completed one full cycle of the Flipped Mess. To run the pattern continuously, make an under-the-arm throw before uncrossing your dominant hand and then simply do another cycle. If you are already skilled at performing Mills Mess, the Flipped Mess will likely just "click" after you practice a cycle or two. Your mind will naturally grasp the shape and rhythm of the pattern and it will therefore be easy to learn.