  For certain contact manifolds admitting a 1-periodic Reeb flow we construct a
conjugation-invariant norm on the universal cover of the contactomorphism
group. With respect to this norm the group admits a quasi-isometric
monomorphism of the real line. The construction involves the partial order on
contactomorphisms and symplectic intersections. This norm descends to a
conjugation-invariant norm on the contactomorphism group. As a counterpoint, we
discuss conditions under which conjugation-invariant norms for
contactomorphisms are necessarily bounded.
