I could be wrong, but weren't ex-Saddam's army soldiers (under leadership of an officer form Saddam's military) sent by themselves to clean out Fallujah in the early years of the occupation and ended up joining the insurgency?
This one is one of those stories that depends on how you want to interpert what you are saying.
Please recall that the kind of people that led the insurgency in Fallujah would be the kind of people that would have shot one of Saddam's officers on sight, war or no war. Old hatreds die hard in that area.
This is about like saying, "Oh, I heard the Black Panthers were rioting in Atlanta, and a Klansman went down and joined up with them." Not impossible, but seems kind of ASB to me, on the face of it.
A lot of the post war Iraqi Police (IP) and Iraqi Army (IA) were associated with US forces who basically fed and clothed them from US and US-captured supplies.
Other IP and IA units were handled directly by Iraqi officials paid by the Transitional Government (or whatever it was called that week -seemed like it was getting a name change every other week for a while) those units might or might not get paid or fed or supplied, depending on what the official felt like doing.
So - your story about Fallujah and IP/IA units deserting and joining the insurgents? In some cases, guys from the non US-associated Units did go down, listen to the insurgents and say the equivalent of "Fukitall, they ain't going to pay me or take care of me, I'm going to the other side."
Can you really blame them?
At least in one case I heard of, there was an insurgent killed, whose name was on the rolls of an IA unit, but had never actually been there - the OIC had padded the rolls with fraudulent names to get more money for himself.
But the IA unit associated with the US forces - the folks who were properly fed, led, and taken care of, did a good job, from everything I heard.
Yeah, they weren't as good as our guys, from what I heard from Marines and Army that were there. Brave in about the same proportion as our guys, just need some better training and equipment, for the most part.