Is it possible to rotate an angle on perfectly smooth ice? You must start and end with the same pose.
Prove it if you think you can't.

 A: Yes, it is possible.  This is an example of the cat-righting problem.
You simply must change your moment of inertia during the process.
Model your body as two cylinders of equal mass which can exert forces between each other to start spinning.  The bottom cylinder has an adjustable radius, initially set equal to the top cylinder's.
Set the top cylinder spinning CW.  To conserve angular momentum, the  bottom cylinder spins CCW at the same angular frequency.  
Wait a short time, then suddenly increase the radius of the bottom cylinder and simultaneously make the top cylinder change directions to start spinning CCW at the same angular frequency as before.  The bottom cylinder will start spinning CW, but since its radius is increased, its moment of inertia is higher, and its angular frequency will be smaller than before.
When the two cylinders are lined up relative to each other, stop them both, and return the bottom cylinder to its original radius.  The entire apparatus must have rotated because the top cylinder's angular frequency had only a one absolute value the entire time, but the bottom cylinder had two.  They therefore had different total angular displacements, so the entire thing must have rotated.
