Directional mud tires will work better in mud when used in the proper direction. The directional tread is a "chevron" pattern, like a tractor tire, and is designed to sweep the mud out of the tread lugs as the tire spins. If you turn the tire around the tread lugs will bite more the first time they dig into the mud, but as they spin the mud is actually packed into the tread pattern instead of being swept out of the tread pattern, and the tire becomes a racing slick because all of the tread lugs are packed full of mud.
Turing a mud tire around backwards can improve traction on surfaces other than mud though. This is primarily the case in sand, because the sand does not pack into the tread lugs. This also can be the case with snow, depending on the snow conditions. Running mud tires backwards in sand and snow can turn them into semi-paddle tires, and they can hook up very good. Unfortunately, they can also dig a grave in a big hurry. Sand tires work better in sand than backwards mounted mud tires do. Snow tires work better in snow than backwards mounted mud tires do. And mud tires work better in mud, when they are run in the proper direction.