  We introduce a pseudo-metric on the contactomorphism group of any contact
manifold $(M,\xi)$ with a cooriented contact structure $\xi$. It is the contact
analogue of a corresponding semi-norm in Hofer's geometry, and on certain
classes of contact manifolds, its lift to the universal cover can be viewed as
a continuous version of the integer valued bi-invariant metric introduced by
Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only
if $(M,\xi)$ is strongly orderable and that its metric topology agrees with the
interval topology introduced by Chernov and Nemirovski. In particular, the
interval topology is Hausdorff whenever it is non-trivial, which answers a
question of Chernov and Nemirovski. We discuss analogous results for isotopy
classes of Legendrians and universal covers.
