Figure: 
: Evolution from a gap 4 pair; no phase is shifted
nor reflected relative to any other.
Some cycles run their course without returning to the same symmetry
class; others may shift the original configuration several times
before returning to their original form. Alternatively, they may pass
through a reflected image before repeating; most cycles of the form
operate by reflection but a shift by half the length of the
ring could produce the same result. Sometimes there is no distinction
between translation and reflection. If N and p are relatively prime,
quite long periods can result.
Amongst the last two examples, all the phases of Figure 
are
distinct, wheres Figure manifests reflective symmetry. Given
the way that boundaries interact, a pair with one gap may evolve into a
pair with another gap, and then evolve back into the original pair,
although this phenomenon is not present on a ring of length sixteen.