  A well-known result by Palamidessi tells us that \pimix (the \pi-calculus
with mixed choice) is more expressive than \pisep (its subset with only
separate choice). The proof of this result argues with their different
expressive power concerning leader election in symmetric networks. Later on,
Gorla offered an arguably simpler proof that, instead of leader election in
symmetric networks, employed the reducibility of incestual processes (mixed
choices that include both enabled senders and receivers for the same channel)
when running two copies in parallel. In both proofs, the role of breaking
(initial) symmetries is more or less apparent. In this paper, we shed more
light on this role by re-proving the above result - based on a proper
formalization of what it means to break symmetries without referring to another
layer of the distinguishing problem domain of leader election. Both Palamidessi
and Gorla rephrased their results by stating that there is no uniform and
reasonable encoding from \pimix into \pisep. We indicate how the respective
proofs can be adapted and exhibit the consequences of varying notions of
uniformity and reasonableness. In each case, the ability to break initial
symmetries turns out to be essential.
