  We consider dual unitary operators and their multi-leg generalizations that
have appeared at various places in the literature. These objects can be related
to multi-party quantum states with special entanglement patterns: the sites are
arranged in a spatially symmetric pattern and the states have maximal
entanglement for all bipartitions that follow from the reflection symmetries of
the given geometry. We consider those cases where the state itself is invariant
with respect to the geometrical symmetry group. The simplest examples are those
dual unitary operators which are also self dual and reflection invariant, but
we also consider the generalizations in the hexagonal, cubic, and octahedral
geometries. We provide a number of constructions and concrete examples for
these objects for various local dimensions. All of our examples can be used to
build quantum cellular automata in 1+1 or 2+1 dimensions, with multiple
equivalent choices for the ``direction of time''.
