Hi Forum,
So I was thinking to myself and teammate and was wondering how important (if at all) is symmetry in your robot? It seems to be that in the past the majority of the great robots have been symmetrical. However, every now and then there is a totally asymmetrical successful robot (400s, for example). My question is, why does this trend exist? Are there actual advantages of symmetry vs asymmetry?
Symmetric robots have a greater ability to drive straight, since they are more balanced. And they look more beautiful 
It’s really all about balance. You brought up 400S’s robot as an example of an unbalanced robot that did a bit well (It only won Worlds. No big deal.). You can re-watch the Worlds finals and 400S’s other matches. You can see that they had to realign a lot (Not much at Finals, but in the qualification matches.) due to the force of their linear puncher and by how light their robot was. I believe that new people just integrate this idea of “symmetrical is necessary” when building anything, but when you look at it really, as long as you can fix yourself in case of an error, it really doesn’t matter what you choose. However, asymmetrical builds generally allow for more features.
So to summarize and answer your questions:
1.) They were taught this way and it is so easy to remember that it sticks.
2.) Weight distribution = balance, but you may pass up some creative possibilities.
I agree that symmetry is often a constraint that teams will unnecessarily impose upon themselves, when really as long as you have a good weight distribution, you should be fine. Our robot is currently very symmetrical, but after analyzing our planned hang mechanism, we realized the optimal placement for the mechanism would be off to one side of the robot instead of right in the center. If you can get a distinct advantage, I would say that sacrificing symmetry is definitely worth it.
In a way, symmetry can be considered a “human instinct”. When building something, we automatically tend to place the parts in a symmetrical fashion without giving much thought to it. And when something is symmetrical, we tend to consider it to be better looking.
In my opinion, there is pretty much no distinct advantage to a completely symmetrical robot in vex, since you can use counterweights (such as extra screws on one side) to achieve balance on an asymmetrical robot.
I think that the base of your robot should probably be semetrical so the movements of your robot arnt super awkward.(excluding kiwi drives)
You need a good center of balance for your robot to drive nicely and it is more important in holonomic drive trains. Whether symmetry or lopping on weight to some part of the robot helps is up to you but building it symmetrically is the easiest way to achieve a nice center of gravity location as projected on the plane of the floor (X&Y not Z).
But a two-wheel drive robot does not need as much symmetry as all the contact points for the driven wheels are in one part of the robot. Left/right symmetry is wanted, but front/back is needed less on those builds. Just don’t let it tip over or do a wheelie (uncontrolled wheelie that is)