I recently discovered that UIViews should only have UIViewControllers when they fill the entire window (or are managed by another UIViewController such as a UINavigationController or UISplitViewController). This quotation is from the documentation for UIViewController:
You should not use view controllers to manage views that fill only a part of their window—that is, only part of the area defined by the application content rectangle. If you want to have an interface composed of several smaller views, embed them all in a single root view and manage that view with your view controller.
I usually put my view logic in the UIView even when it is managed by a UIViewController, yet I often find myself needing to access properties of the UIViewController such as its navigationController property. However, UIViews are not supposed to be aware of their UIViewController.
My conclusion is that view logic should go in a UIView's UIViewController when one exists, and in the UIView itself otherwise.
Alternatively, is it better practice to create a controller class for a view which is not a subclass of UIViewController? UIPopoverController (an NSObject subclass) appears to follow this pattern, although in most cases (UIButton, etc.) views do not seem to have dedicated controller classes.