is there a way to change the order of the cast/crew credits (I.e. moving produced by, cinematography by, etc) around?

Also, bonus question, can I reorganize the credits underneath a specific category?

Jarrale: The only credits that can be ordered based on their on-screen order are the Cast and Writers. In all other departments, IMDb can only list people in alphabetical order.

It's not possible to re-arrange the departments relative to each other for a particular title; the order is fixed.