We show how the metric, the almost complex structure and the almost product structure of the homogeneous nearly Kahler S3 × S3 can be recovered from a submersion π : S3 × S3 × 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.