The process described in "Chromosome to cycle1" is in fact invertible, as described by the following pseudocode.
CycleToChromosome(N)
for j ← 1 to |N| / 2
if n2j - 1 < n2j
cj ← n2j / 2
else
cj ← −n2j − 1 / 2
return C
>>> Cycle(1, 2, 4, 3, 6, 5, 7, 8).to_chromosome() Chromosome(1, -2, -3, 4)