  In this paper, we describe a new surprising example of a fibration of the
Clifford torus S3 x S3 in the 7-sphere by great 3-spheres, which is fiberwise
homogeneous but whose fibers are not parallel to one another. In particular it
is not part of a Hopf fibration.
  A fibration is fiberwise homogeneous when for any two fibers there is an
isometry of the total space taking fibers to fibers and taking the first given
fiber to the second one.
  We also describe in detail the geometry of this surprising fibration and how
it differs from the Hopf fibration.
