Direction of arrows in electrocyclic reactions
The answer is rather bland and probably not what you want to hear. Electrocyclic reactions are called electrocyclic because they are cyclic. (Please don’t see cyclic circular reasoning in there.) They proceed in a cyclic, i.e. fully symmetric fashion. You simply cannot draw one arrow direction and claim it is the most logical or the only one possible. You could draw these arrows as you drew them, but you could also formulate a radical-based mechanism (each double bond splits into two radicals which combine with the two of the other double bond) at no loss.

Thus, there is no way to say whether left or right. The reaction happens; in a cyclic fashion.