Is there a specific way PCAs are meant to be done? It seems as though the idea of the PCA is to try as many different ways as possible until you get something reasonable out of it. Then make post-hoc explanations as to why you did it in that particular way. This does not seem like a satisfactory method to go about it.

Why would you use, for example, an equamax, varimax or quartimax rotation? Are there any specific methods for determing what style of rotation you should use, other than it works?

My expection is that all of the different methods should give you the same results as the principal components have a precise mathematical meaning: the first PC is the dimension that maximizes the variance, the second PC is the dimension that maximizes the variance once the first PC is subtracted away (and thus made orthogonal to it), etc...

It's possible some methods are more efficient than others, or more robust.