Ideally, I think they should be equivalent and both should require buy-in from the other on major decisions. Making management decisions in a tech company without intimate knowledge of what can and cannot be done, and in specific time-frames is ridiculous (in the cases where the decision relies on engineering), just as making technical decisions without intimate knowledge of what the market wants, needs, and will respond to is ridiculous (in the cases where it affects products in the market).

> I wager there are at least some decisions a PE needs Director approval on (where the reverse is never true).

While a PE is probably not /asking/ for Director approval on their decisions, I would say they do have the power to create almost as much business impact by making good or bad decisions in their work -- both formal decisions about things like architecture, stack choices, etc as well as informal decisions like how they treat their teammates, who they recommend for hire, etc. The old trope of "the old greybeard in the corner who won't let us do anything" comes to mind.