OK, the center of pressure concept was dropped some years back due to a pesky matter of the Cp being located at infinity behind the trailing edge when you run the airfoil at the zero lift angle. The current method of describing the twisting action is that the center of lift is located at the 25% point of the airfoil and that any twisting force is called the pitching moment.

And with any airfoil it won't matter if you operate it backwards or not the center of lift will always be at the 25% point behind whichever end is the leading edge of the moment. What would change is the pitching moment since it's largely related to the camber value.

You're not going to find any airfoil that is optimized for both directions. Low drag requires a sharp trailing edge. But to soften the stall and avoid other issues you want a rounded leading edge. The one area where this alters and a truely symetrical airfoil would work is at very, very low Reynolds numbers like with indoor or slow flying free flight models where a simple arc curved flat plate works fine and is the airfoil of choice like with indoor models. But for general RC flying and dispite Tall Paul putting his wing on backwards as a giggle it's generally not a great idea.