  We study pairs of Engel structures on four-manifolds whose intersection has
constant rank one and which define the same even contact structure, but induce
different orientations on it. We establish a correspondence between such pairs
of Engel structures and a class of weakly hyperbolic flows. This correspondence
is analogous to the correspondence between bi-contact structures and
projectively or conformally Anosov flows on three-manifolds found by
Eliashberg--Thurston and by Mitsumatsu.
