  We exhibit the first examples of exotic contactomorphisms with infinite order
as elements of the contact mapping class group. These are given by certain Dehn
twists on the separating sphere in a connected sum of two closed contact
3-manifolds. The tools used to detect these combine an invariant for families
of contact structures which generalises the Kronheimer-Mrowka contact invariant
in monopole Floer homology, together with an h-principle for families of convex
spheres in tight contact 3-manifolds. As a further application, we also exhibit
some exotic 1-parametric phenomena in overtwisted contact 3-manifolds.
