  We prove a rigidity result for group actions on the line whose elements have
what we call "hyperbolic-like" dynamics. Using this, we give a spectral
rigidity theorem for $\mathbb{R}$-covered Anosov flows on 3-manifolds,
characterizing orbit equivalent flows in terms of the elements of the
fundamental group represented by periodic orbits. As consequences of this, we
give an efficient criterion to determine the isotopy classes of self orbit
equivalences of $\mathbb{R}$-covered Anosov flows, and prove finiteness of
contact Anosov flows on any given manifold.
  In the appendix with Jonathan Bowden, we prove that orbit equivalences of
contact Anosov flows correspond exactly to isomorphisms of the associated
contact structures. This gives a powerful tool to translate results on Anosov
flows to contact geometry and vice versa. We illustrate its use by giving two
new results in contact geometry: the existence of manifolds with arbitrarily
many distinct Anosov contact structures, answering a question of
Foulon--Hasselblatt--Vaugon, and a virtual description of the group of contact
transformations of a contact Anosov structure, generalizing a result of Giroux
and Massot.
