These diffeomorphisms exhibit weaker forms of hyperbolicity, which are still extremely common, while at the same time very robusts. We study these in dimension 3 and prove some rigidity or classification results. We assume that the diffeomorphism is homotopic to the identity, and show that certain invariant foliations associated with the diffeomorphism have a structure that is well determined. This has some important consequences when the manifold is either hyperbolic or Seifert: under certain conditions we prove the diffeomorphism is up to iterates and finite covers, leaf conjugate to the time one map of a topological Anosov flow. Some of the main techniques used are the use of (branching) two dimensional foliations in 3-manifolds. This is joint work with Thomas Barthelme, Steven Frankel, and Rafael Potrie.