Why does this object periodically turn itself? See below gif image taken from here.

Or see this Youtube video about 30 sec in. 


*

*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?
 A: 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.
http://www.youtube.com/watch?v=L2o9eBl_Gzw
This seems to be a home experiment where a guy throws the spinning object upwards.
http://www.youtube.com/watch?v=3VwS5ykAUHI
And this seems to be a computer simulation.
http://www.youtube.com/watch?v=LR5hkgfRPno
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:
Mark S. Ashbaugh, Carmen C. Chicone and Richard H. Cushman, The Twisting Tennis Racket, Journal of Dynamics and Differential Equations, Volume 3, Number 1, 67-85 (1991). (One time found at http://math.ucalgary.ca/files/publications/cushman/tennis.pdf which is no longer a working link.)
http://www.youtube.com/watch?v=4dqCQqI-Gis
