Shift-invariant bijections

Let GF(2) denote the field of two elements.
Fix a natural number n, set V=GF(2)^n, and let S be the shift-operator on V.
A function F from V to V is called shift-invariant if FS=SF and there is a one-to-one-correspondence between such functions and Boolean functions on n variables.
We discuss various problems related to finding and studying shift-invariant functions that are bijective, and their relevance to cryptography.