  Random quantum circuits continue to inspire a wide range of applications in
quantum information science and many-body quantum physics, while remaining
analytically tractable through probabilistic methods. Motivated by an interest
in deterministic circuits with similar applications, we construct classes of
\textit{nonrandom} unitary Clifford circuits by imposing translation invariance
in both time and space. Further imposing dual-unitarity, our circuits
effectively become crystalline spacetime lattices whose vertices are SWAP or
iSWAP two-qubit gates and whose edges may contain one-qubit gates. One can then
require invariance under (subgroups of) the crystal's point group. Working on
the square and kagome lattices, we use the formalism of Clifford quantum
cellular automata to describe operator spreading, entanglement generation, and
recurrence times of these circuits. A full classification on the square lattice
reveals, of particular interest, a "nonfractal good scrambling class" with
dense operator spreading that generates codes with linear contiguous code
distance and high performance under erasure errors at the end of the circuit.
We also break unitarity by adding spacetime-translation-invariant measurements
and find a class of such circuits with fractal dynamics.
