Remember it's de facto "Kurdistan", not de jure. Knowing the difference is part of what makes the Kurds smart.

Why they volunteer for Baghdad duty--no doubt the pay--by the US--is good. They are also probably the best for the job--seen as not having a dog in the Sunni-Shiite fight. On the other hand they can be seen as mercenaries.   

And the more favors they do for Maliki, the less he can oppose their continued-- unofficial-- separation from Iraq.

We'll see how it goes.