No, neither of you seem to really understand how most ORs are run.
What the anesthesiologists do is up to them. They can wear cloth, non-disposable scrub caps that they brought in from the outside, because they're not standing next to the patient during the operation. They're either at the patient's feet (in an ENT case) or behind a sterile drape.
Just because the anesthesiologist's newspaper "isn't sterile" doesn't mean squat for you, the med student on your surgery rotation, or for you, the surgical resident. He's hiding behind a sterile drape - sterility doesn't matter. But you're on the sterile side of the drape, and it DOES matter.
What the surgeons wear DOES matter. If you're operating, you should not wear a cloth cap. If you do, it should be covered by a disposable cap that is provided by the hospital.
IF YOU ARE ENTERING THE OR, YOU SHOULD NOT - I repeat, NOT - BE WEARING A LONG-SLEEVED SHIRT UNDER YOUR SCRUBS!!! It is NOT sterile technique, and you will get thrown out of the OR to go change. Med students have gotten thrown out for wearing t-shirts/tank tops under their scrubs.
YOU SHOULD ALSO NOT WEAR SCRUBS STOLEN FROM AN OUTSIDE HOSPITAL. Hospitals issue scrubs that are only one color. Anybody wearing scrubs of a different color (i.e. were clearly gotten from outside the hospital) will be told to leave the OR and go change - no exceptions. Not even for attendings!
There are no such things as "sterile scrubs." Yes, you technically cover everything up with a sterile gown, but still, scrubs should not be taken out of the hospital. No, I know that everyone takes their scrubs outside of the hospital. But what your resident can get away with, and what you can get away with, are 2 different things.
OP - I honestly have no idea what they would do about someone who insisted (for religious reasons) on wearing a scarf in the OR. What would most likely happen is that the charge nurse would have to contact the company that provides the hospital with scrubs. That company would have to provide head coverings (made of the same material as the scrubs) for that individual. That would be the only head covering "allowed" in the OR. I'm guessing.