The first time I saw this (was on Digg a few months ago) I spent about ten minutes looking at it, and figured out how to change the direction at will. I found that looking at a spot that doesn't move much (i.e. the leg not pointed outwards) and blinking works. You might need to do it a few times, or keep changing the spot you're looking at, but it should do it eventually.
I saw clockwise first, and then changed it to anti.
Do this, and more than likely, the animation will change direction every time: Place your hand on the monitor, or very close to it, with your fingers (at the tips) about a 1/4 inch apart, to where you can see the dancer through them but not completely. Every time you put your hand up, then remove it, you should see the dancer spinning in the opposite direction.
This only reason this tricks your brain is because there is no shading, lighting or texture on the model, therefore there is no way for the human brain to register the movement by depth perception. This is when either the left or the right side of the brain decides to take over and make a wild guess.
With some people, the left and right side of the brain will literally fight over which will comprehend this trickery and thats why sometimes you see it one way, and sometimes the other.
If you look at the dancer, you'll notice that spinning clockwise, her position looks natural for that direction. When she seems to be spinning counter-clockwise, you'll notice that her body positioning (arms especially) look unnatural, like shes spinning backwards.
What causes you to think shes spinning clockwise is naturally the position of her arms. What causes you to think shes spinning counter-clockwise is when you notice the shadow is always spinning counter-clockwise, and that it doesn't quite fit the clockwise motion.
So the trick here is to design a model that looks natural spinning one direction (clockwise) and your right brain calculates this, but putting an element such as her shadow that spins in the opposite direction of what her natural pose represents, then causes your left brain to kick in because it seems to be the logical thing to do regardless of her pose.
Heh, I only changed the direction like once. As it is it took quite a while to get it to work. I had gotten bored and given up a few times before it changed, and after it did I gave up on changing it again.
However the boys that linked it were amused for an extremely long time. Tell her to try again, and be patient. I'm betting everyone can change the directions.
do this: make some math (simple if you want like 2+58, 49-13) then you will watch it counter clockwise
then take a pencil and a paper and do some sketches, don't force yourself just do whatever your imagination is able to do in that moment (a house, some circles, a sun, etc) and you will see the image clockwise.
I started clockwise. I tried to get it to switch by covering most of the image save the side. It didn't work at first, that is, until I tried the other side. Then I could switch it at will. Crazy. I love stuff like this.