  We study Floquet systems with translationally invariant nearest-neighbor
2-site gates. Depending on the order in which the gates are applied on an
N-site system with periodic boundary conditions, there are factorially many
different circuit configurations. We prove that there are only N-1 different
spectrally equivalent classes which can be viewed either as a generalization of
the brick-wall or of the staircase configuration. Every class, characterized by
two integers, has a nontrivial space-time symmetry with important implications
for the level-spacing distribution -- a standard indicator of quantum chaos.
Namely, in order to study chaoticity one should not look at eigenphases of the
Floquet propagator itself, but rather at the spectrum of an appropriate root of
the propagator.
