  Optical continuous-variable cluster states (CVCSs) in combination with
Gottesman-Kitaev-Preskill~(GKP) qubits enable fault-tolerant quantum
computation so long as these resources are of high enough quality. Previous
studies concluded that a particular CVCS, the quad rail lattice~(QRL), exhibits
lower GKP gate-error rate than others do. We show in this work that many other
experimentally accessible CVCSs also achieve this level of performance by
identifying operational equivalences to the QRL. Under this equivalence, the
GKP Clifford gate set for each CVCS maps straightforwardly from that of the
QRL, inheriting its noise properties. Furthermore, each cluster state has at
its heart a balanced four-splitter -- the four-mode extension to a balanced
beam splitter. We classify all four-splitters, show they form a single
equivalence class under SWAP and parity operators, and we give a construction
of any four-splitter with linear optics, thus extending the toolbox for
theoretical and experimental cluster-state design and analysis.
