We show how the metric, the almost complex structure and the almost product structure of the homogeneous nearly Kähler S³ × S³ can be recovered from a submersion π : S³ × S³ 3 S³ → S³ × S³. On S³ × S³ × S³ we have the maps obtained either by changing two coordinates, or by cyclic permutations. We show that these maps project to maps from S³ × S³ to S³ × S³ and we investigate their behavior.