  We explore equivariant dynamics under the symmetric group $S_N$ of all
permutations of $N$ elements. Specifically we study one-parameter vector
fields, up to cubic order, which commute with the standard real
$(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the
linearization at the trivial 1-cluster equilibrium of total synchrony.
  All equilibria are cluster solutions involving up to three clusters. The
resulting global dynamics is of gradient type: all bounded solutions are
cluster equilibria and heteroclinic orbits between them. In the limit of large
$N$, we present a detailed analysis of the web of heteroclinic orbits among the
plethora of 2-cluster equilibria. Our focus is on the global dynamics of
3-cluster solutions with one rebel cluster of small size. These solutions
describe slow relative growth and decay of 2-cluster states. For
$N\rightarrow\infty$, the limiting heteroclinic web defines an integrable
\emph{rebel flow} in the space of 2-cluster equilibrium configurations. We
identify and study the seven qualitatively distinct global rebel flows which
arise in this setting.
  Applications include oscillators with all-to-all coupling, and
electrochemistry. For illustration we consider synchronization clusters among
$N$ complex Stuart-Landau oscillators with complex linear global coupling.
