  Many-body quantum dynamics defined on a spatial lattice and in discrete time
-- either as stroboscopic Floquet systems or quantum circuits -- has been an
active area of research for several years. Being discrete in space and time, a
natural question arises: when can such a model be viewed as evolving unitarily
in space as well as in time? Models with this property, which sometimes goes by
the name space-time duality, have been shown to have a number of interesting
features related to entanglement growth and correlations. One natural way in
which the property arises in the context of (brickwork) quantum circuits is by
choosing dual unitary gates: two site operators that are unitary in both the
space and time directions. We introduce a class of models with $q$ states per
site, defined on the square lattice by a complex partition function and
paremeterized in terms of $q\times q$ Hadamard matrices, that have the property
of space-time duality. These may interpreted as particular dual unitary
circuits or stroboscopically evolving systems, and generalize the well studied
self-dual kicked Ising model. We explore the operator dynamics in the case of
Clifford circuits, making connections to Clifford cellular automata [J. Math.
Phys. 49, 112104 (2008)] and in the $q\to\infty$ limit to the classical
spatiotemporal cat model of many body chaos [Nonlinearity 34, 2800 (2021)]. We
establish integrability and the corresponding conserved charges for a large
subfamily and show how the long-range entanglement protocol discussed in the
recent paper [Phys. Rev. B 105, 144306 (2022)] can be reinterpreted in purely
graphical terms and directly applied here.
