See this video about 30 sec in. http://www.youtube.com/watch?v=dL6Pt1O_gSE
Is this a real effect? Why does it seem to turn periodically? Can it be explained by classical mechanics alone? Is there a simple equation that models this behaviour?
It's a classical mechanics effect for sure although a really interesting one. Following links on "Dzhanibekov effect" one gets at Marsden and Ratiu's "Introduction to Mechanics and Symmetry" Chapter 15 Section 15.9 "Rigid Body Stability" treating this with use of the Casimir functions.
From remark 1: A rigid body tossed about its middle axis will undergo an interesting half twist when the opposite saddle point is reached.
Here is another and more profound example under weightless conditions.
This seems to be a home experiment where a guy throws the spinning object upwards.
And this seems to be a computer simulation.
There is a related unstable orbit effect which you can try out easily yourself with a tennis racket. A treatment due to Ashbauch Chicone and Cushman is here: