Why does Earth rotate on a single axis? You often see gymnasts flip and twist at the same time. They rotate on two axis.
Earth twists, but doesn't flip. It only rotates on one axis. I'm not sure, but I think the same is true for all other planets.
Why is that?
 A: 
You often see gymnasts flip and twist at the same time. They rotate on two axis.

If gymnasts were rigid bodies, that perception of rotating on two axes would just be an illusion resulting from using a non-optimal set of axes. At any point in time, the rotation of a rigid body in three dimensional space can always be described as being about a single axis. Always. This is one of the many consequences of Euler's rotation theorem.
Gymnasts (along with divers, ski aerialists, and others who use acrobatic skills) aren't rigid bodies. They instead obey "falling cat physics", as portrayed below.

(source: physlink.com)
Properly describing the physics of a falling cat has been an interesting problem for over half a century. This problem has attracted the attention of roboticists such as T.R. Kane (the father of modern robotics), mathematicians, and physicists.


Earth twists, but doesn't flip. It only rotates on one axis.

While the Earth isn't a cat, it is subject to external torques. These torques slowly (very slowly, over the course of 26000 years) make the Earth's rotation axis precess. Polaris has not always been and will not always be the North Star. In about 13000 years, the much brighter star Vega will be the North Star rather than Polaris. In another 13000 years after that, Polaris will once again be the North Star.
The Earth is also conjectured to have undergone episodic true polar wander (pdf link) at points in the past. While not quite cat physics, true polar wander is significantly more catlike than is precession/nutation.
A: The short answer is: Conservation of angular momentum.
Imagine the velocity of all the points of a sphere (planet) at any point in time: There is some axis at rest (the rotation axis) and all the other points are moving around this axis. Now, with everything being symmetrical, why would that axis change over time?
In more technical terms: In order for the angular momentum of the earth's rotation to change (which "flipping" would mean), there would have to be some external torque on it.
Now, for bodies with less symmetry, their movement is more complicated if the angular momentum is not aligned with an axis of symmetry (if there is any).  (Sadly, gymnasts are neither spheres nor rigid.) I find it hard to "simulate" this in my head, that's why we use maths for these things. However. conservation of angular momentum still applies, which means that the kind of independent twisting and flipping you imagined (as I understand it) is not possible for any rigid body.
A: The axix of earth's rotation was determined during the process of earth's formation, and later by other impacts (mainly the one that may have formed moon). Movements along any other axis (if any) would have been cancelled during formation process. After that there is no way for earth to change that axis by itself. Even if it happened due to some event, the rotation would still stabilize to a single axis just like it did during the formation process.
Any attempt to make it rotate on two axis would basically extend the formation process in a way.
