1stGenCRXer

I don't think anyone has quite nailed this answer on the head yet, so I'll contribute.

Crossing over hands when turning the wheel does two things. First, it means you have one hand off the wheel in the middle of a turn [possibility of a hand slipping and losing your wheel position in the middle of the corner]. And second, it means that the leverage of your arms on the wheel will change as you grab, release, etc, which will make your turns choppy.

Instead of crossing over in a competition situation, what you should do is position your hands before you get to the turn so that in the middle of the turn your hands will be relatively close to 3 and 9 o'clock on the wheel, so that you can more smoothly make corrections to your wheel input at the apex of the corner. This will prevent awkward positioning of your arms at the most critical part of the corner, and also ensure that you have a firm grasp on the wheel.

Granted that in a manual transmission car, you spend a lot of time with only one hand on the wheel, but ideally your upshifting and downshifting should be executed in a straight line. This way, as you approach your corner entry, you can have your left hand in position and quickly grasp the wheel with your right hand after getting into your desired corner exit gear, and still be able to smoothly transition into the corner with both hands in the wheel, without putting your arms in an awkward position.

The key thing to remember is that if your arms start to feel out of place, you won't be able to judge as smoothly on your correction inputs. This is more true for street cars than dedicated race machines that can have variable rate steering [such as Formula 1, where they steering is set tight enough for the tightest turn on the track and no more, meaning that drivers do the "thumb hook and cross" motion]. Set up for the corner ahead of time, be smooth on the wheel, and I think you'll find the technique more coordinated than trying to flip hand-over-hand for rapid corners.