I am not nay-saying your hypothesis, but want to provide another perspective... When I draw a spiral, I tend to start in the middle and work my way out, so maybe the direction could be reversed. Also, if these are Ulam spirals, they would be starting in the middle working their way outwards.

I do agree that they seem to indicate a direction to follow for the order the information in the flames (if it exists). Following this reasoning, out of the 4 spirals, there are 2 directions of travel. I instinctively tend to lean towards one direction for the inner set and the opposite direction for the outer set.