I'd have to say that something's wrong...it shouldn't spray off-center.
By indicating that the direction of spray doesn't change when you rotate the needle would suggest that the needle is not the problem...perhaps there is a machining error in the hole that the needle passes through? Even a minute burr would deflect paint in a uniform way that wouldn't be influenced by your needle's rotation. Other than that, I'm stumped....good luck
Cheers