Time-frequency shift invariance of Gabor spaces generated by integer lattices

We study extra time-frequency shift invariance properties of Gabor spaces. For a Gabor space generated by an integer lattice, we state and prove several characterizations for its time-frequency shift invariance with respect to a finer integer lattice. Some extreme cases are also considered. The result obtained shows a close analogy with the extra translation invariance of shift-invariant spaces, however, presents subtle but deep differences, due to the non-commutativity of the time-frequency operations.