Yet another idea:

some leaves are pointing inside chess squares,
what if they indicate the reading order of  flames sides and the direction.
Direction could be imposed by the corner spirals:
if a leaf is pointing  left we go CCW, otherwise we go CW.

https://imgur.com/oLzQaRP

unfortunately I'm not sure if the green color is correct (might also be "inner top" flames)
and if the black color should be taken into account, if not we would have this order (based on leaf marked yellow):
1. inner right CCW
2. outer bottom CW
3. outer top CW
4. inner left CW