The most common outline of a surfboard that comes to people minds would be a basic outline with symmetrical rails leading to the meeting points at the nose and tail. My question to challenge this idea would most likely be, "Does the water really know, or care if the rails are symmetrical?" We ourselves, from back to front, are not symmetrical beings. Looking down at one's toes and then looking behind your heels you will see a bit of a difference. How does this translate to board design and does it make a difference in the water?
When we walk, run, surf, or move, our bodies have been designed to do these things in a manner in which one direction has an advantage over the other in that particular flow of action. Running down a hill backwards is very difficult, but running up a hill backwards can actually be very efficient. Weight placement and leaning off one's center of gravity creates force or leverage towards a particular direction. With surfing, the idea behind turning is to create a directional change while being able to maintain balance and control. This is most easily accomplished by staying over or close to the center of gravity above your board.
When I ride asymmetrical boards, I am most likely to hear from others how great it would go in one particular direction. This is a common misunderstanding of asymmetrical design. The whole idea behind having asymmetry in the rails is to harness the best parts of a board and combine it into one board that functions smoothly and efficiently in both directions.
I surf with my right foot forward. So the idea is to have a straighter rail curve on my toe side (the side of the board that my toes are facing when I stand). This produces a more effective rail to generate speed and control down the line. The opposite rail or heel side would be curved more and sometimes shorter, leading to a smaller portion on the rail in the face of the wave at any given time, enabling an easier pivot point to be created.
The flow of water makes a difference. Water coming up the face of the wave is a force just as important to consider as the force of water passing under your board as you travel forward. On a wave with power, these forces are much more evident than compared to a weak, small wave. The energy produced by the wave or by your weight placement needs to be displaced on the water surface in a manner that is able to create a controlled amount of directional speed.
With the combination of rail curves and bottom board contours (which are asymmetrical as well) these areas of board control are more easily felt and maneuvered without passing over the center of gravity as far as one would go on a symmetrical board.
The silly side of the story would be that people have surfed and will surf on a massive array of surf craft and do incredible, amazing things. Asymmetrical design is an area of surfboard development in which I believe there is much more ground to traverse, but is not the only or absolute "answer" to surfboards. All together, they are super fun to ride, extremely functionional, and pretty amusing to look at.