  We demonstrate that the dynamics of kicked spin chains possess a remarkable
duality property. The trace of the unitary evolution operator for $N$ spins at
time $T$ is related to one of a non-unitary evolution operator for $T$ spins at
time $N$. We investigate the spectrum of this dual operator with a focus on the
different parameter regimes (chaotic, regular) of the spin chain. We present
applications of this duality relation to spectral statistics in an accompanying
paper.
