I'm not sure what "shaded symmetry" is, but I can assure you that there is no sideslip in a crab. A crab is basically similar to the canoe situation stated earlier. If you wanted to track straight across a river, you'd have to point somewhat upstream to counteract the current, but you don't have to keep exerting force to keep the bow pointed that way.
I'm not sure why the subject got so confused over the years with forward slips, side slips, skids, etc., etc. It's all the same thing. If you simply think of the relative wind as a vector, it's direction relative to the aircraft axes can be described with two angles. Alpha (or angle of attack) and beta (angle of sideslip). Roll doesn't factor into it, so all the minor distinctions between all those terms above are essentially meaningless when it comes to how the aircraft is going to respond aerodynamically.