How do objects change their axis of rotation? If I hold a pencil at its end and spin it, throwing it upwards, it will spin about its end, but will soon start spinning around its center. How is this?
I would draw the following torque diagram for while it's in the air:


*

*Object: uniform thin rod with length $\ell$ and moments $I_{center}=\frac 1 {12} m\ell^2$ and $I_{end}=\frac 1 3 m\ell^2$)

*Center of rotation some small distance $d$ from the end

*Torque $m\vec g$ downward, at center of mass, with $\theta = 90°$ and $r = \frac \ell 2 - d$

*Possibly wind resistance $\vec D$ upward, at center of mass, with $\theta = -90°$ and $r = \frac \ell 2 - d$


Thus, $\vec \tau = \sum {\vec r \times \vec F} = \sum {rF~sin\theta} = \left (\frac \ell 2 - d\right)(m\vec g - \vec D)$. I could see how this might cause it to spin, but how does the center of rotation to move?
EDIT: Here's a picture because apparently I wasn't clear. It definitely is spinning around the end (or close to it, anyway) before I release the pencil.

 A: This looks like an example of the Tennis Racket Theorem.  Some axes of rotation for a rigid body are more stable than others. If the initial rotation axis does not correspond to one of the principal axes, a wobble can grow and cause the rotation axis to move to a principal axis. This is a result of Euler's Equations of Motion and the moments of inertia.
The tennis racket theorem is a result in classical mechanics describing movement of a rigid body with three distinct angular momenta. 
A: Not sure if I am interpreting your description of the problem correctly, but if I take the initial conditions (using nice round numbers) as a rod of $r = 1\:\mathrm{m}$ length, pivoting about one end at at $\omega = 1\:\mathrm{rad/s}$ at $t=0$, I can decompose the motion as a COM motion of $v=r\,\omega/2$ with a spin about the COM of $\omega$. If I plot the motion (ignoring gravity), I get:

where $t=0$ is the horizontal line at the bottom and the "pencil" is moving upwards after release. While the initial conditions are set by the spin around the end, the following motion is still mechanics about the COM. 
