  Garbe, Hladk\'y, \v{S}ileikis and Skerman recently introduced a general class
of random graph processes called flip processes and proved that the typical
evolution of these discrete-time random graph processes correspond to certain
continuous-time deterministic graphon trajectories. We obtain a complete
characterization of the equivalence classes of flip process rules with the same
graphon trajectories. As an application, we characterize the flip process rules
which are unique in their equivalence classes. These include several natural
families of rules such as the complementing rules, the component completion
rules, the extremist rules, and the clique removal rules.
