There's a video making the rounds about how hard it is to re-learn how to ride a bicycle if you reverse the control sense of the handlebars. It's an interesting video, but I think it misses a very important point.
I need to digress for a disclaimer: I have not had the opportunity to ride a reverse-handlebar bicycle, so I have not put the theory I am going to advance here to the test. If anyone knows where I can find such a bike in the Bay Area I would really welcome the opportunity.
The reason people have a hard time learning to ride a bicycle for the first time is that they think that the direction of the bicycle is controlled by turning the handlebars the way that the direction of a car is controlled by turning the steering wheel. It's not. The mechanics of two-wheel vehicles are completely different from four-wheel (or even three-wheel) vehicles. A bicycle's direction is controlled not by turning the handlebars, but by shifting your weight.
There's a simple experiment you can do on an ordinary bicycle to convince yourself that at least part of what I have just said is true: find a moderately sloped hill that lets you coast at a moderate speed (10-15 MPH) without pedaling. Stabilize your trajectory in a more or less straight line. Now let go of one handlebar, so you are "steering" with only one hand. Use that hand to push that handlebar forward. Don't do it abruptly, just apply gentle, even pressure. If you're using your right hand, you will be turning (at least feel like you're trying to turn) your front wheel to the left. What you will find is that your bike will actually turn to the right. If you want to turn back to the left, you have to pull back with your right hand as if you were turning your front wheel to the right. (I say "as if" because you will find that the front wheel actually does turn to the left despite the fact that you are applying pressure to turn it to the right!)
All this only works if you're going fast enough. When a bike is moving slowly, its direction is controlled more by how you turn the handlebars than how you lean. The hardest part of learning to ride a bicycle is learning to manage the transition between these two control regimes. This is the reason that training wheels are worse than useless when learning to ride a bike. They change the bike from a two-wheel vehicle to a four-wheel vehicle, which doesn't undergo this change in dynamics. If you want your kid to learn to ride a bike, take the pedals off instead of using training wheels.
When a bike is traveling at speed, what happens when you apply pressure to the handlebars is this: let's say that you apply forward pressure with your right hand so that the wheel would ordinarily turn to the left. For a fraction of a second, it actually does turn to the left, and the track of your tires moves to the left. But your body is still moving in the same direction it was before, so you have essentially shifted your weight to the right. So at the moment, you are out of balance.
What happens next is a little tricky to describe. Notice that the head tube (the part of the frame that the front fork is attached to, is angled so that the bottom of the tube is further forward than the top. The result of this is that when the bike leans in one direction, the weight of the bike and its rider causes the front wheel to turn in the same direction as the lean. It is this force that controls where the front wheel is pointing when the bike is moving at speed.
The reason a bike is stable when it is moving is not the gyroscopic stability of the wheels, it is the angle of the head tube, which causes the front wheel to want to point in the same direction as the bike is leaning. As soon as you start to lean one way or the other, the front wheel naturally turns in the same direction, which moves the wheels back underneath your center of gravity and "undoes" the lean. In order to turn, you have to intentionally overcome the bike's natural stability and induce a lean in order to force the front wheel to turn to one side or the other.
So what happens when you apply pressure to the handlebars at speed is not that you are turning the bike, but you are inducing a lean. You can do exactly the same thing by actually leaning, and it doesn't take much. Once you are stabilized, just moving your head from one side to the other can be enough to cause your bike to turn.
Once you realize this, it is easy to learn to ride without having your hands on the handlebars at all. You slowly release your grip until you just have your fingertips on the handlebar. At this point you will notice that you can control your bike by applying pressure to the sides of the seat of the bike with your inner thighs, or by tilting your head back and forth. It takes just a few minutes to learn how to steer the bike this way, at which point you can just take your hands completely off the handlebars. At that point, of course, it doesn't matter whether the handlebars are reversed or not.
But, of course, all this only works once the bike is moving. The hard part is getting to that point from a standing start.
The key here (and at this point I'm really only guessing) is to remember two things: 1) the object of the game is to get the bike moving as quickly as possible and 2) what you're trying to do during that time is not to steer, but simply to keep the front wheel straight. To do that with a reverse-sense handlebar you do have to change your mindset. My guess is that what would work best is to make a conscious effort to think of the process as a game of "chase the front wheel with the handlebars", i.e. if you see the front wheel turning to the right, you "chase" it by turning the handlebars to the right, and vice versa. The result will be the wheel wobbling back and forth around its forward position, but that should be enough to keep you upright long enough to get up to speed.
My prediction is that someone who has read this blog post can learn how to ride a reverse-handlebar bike much more quickly than someone who hasn't. I'll bet that I can learn to ride such a bike in an hour if I had an open space free of obstacles to practice in (like a parking lot). The reason I would need this is that initially I am not going to be able to control the direction of the bike, just keep it stable long enough to get up to speed and into the stable control regime. Like I said in the opening, if anyone knows where I can get my hands on such a bike so I can do this experiment please let me know.