I'm studying different tests for uniform random number generators. I've read about the diehard tests and TestU01. These tests test different properties of a uniform random number generator. The birthday test for instance, tests the uniformity of a random number generator, while the permutation test checks if values are correlated (if I understand it correctly).
Question: Is it possible to assign different tests for RNGs into different categories? And if so; which categories are there? Are there just uniformity and iid-ness?