I have to admit I had that thought - if I needed to get down a narrow staircase in a hurry, I'd prefer if it were anti-clockwise.
But apparently another competing theory is that clockwise staircases allow people to put their right hand against the wall for balance/safety... implying if they had anything to carry (including a lamp!) they'd prefer to do so in their left hand, which isn't too convincing either. But in fact on further consideration, the fact that if you have something largish to carry, you'd probably want it on the wider side, and would be more likely you'd carry it in your right hand, might have something going for it...though I'm not sure why that would be obviously more so going down than going up (and logically I'd expect more things would be carried up from the ground floor than v/v, particularly shortly after building the castle. From the upper floor you can discard used/broken objects by tossing them over the side!).
I don't think going down was a priority. When you are under attack, you want your archers to rush up to the walls, with some extra fighters for good measure, and you don't want them to go down until the enemy has decided to leave. At that point, the speed of your archers going down doesn't matter much.
Hmm...most archers would be right handed and used to holding/carrying their bows in their left hand (during battle at least), which I assume would indeed be easier ascending a clockwise staircase...wonder how you'd test that theory!
I wonder what the etiquette for oncoming traffic was, those spiral staircases with a central spine aren't really walkable anywhere but at the outer wall.
Whoever has to step to the spine side would probably want a hand on the spine, palm making contact from the uphill side no matter wether facing up or down. So for someone uphill, the spine contact would be made with the outside hand (arm crossing in front), for someone downhill with the inside hand. On a clockwise staircase, this would leave the right hand comfortably idle for candle, tool or whatever the person deferring to (presumably higher ranking?) oncoming traffic on the wall side is carrying.
