Well, this sectarianism is oftentimes overplayed. I would say that if you look at the economic incentives, the widespread corruption, and the fact that many of these people thought that their job in the army was a sinecure, you get much closer to the actual reasons for them to not pick up weapons and fight.
Even someone as jaded as I am, who has been looking at this for a long time, was very surprised to notice the fact that they didn't even bother to sabotage the equipment, so that equipment was then quickly turned around and utilized against the Iraqi army.