We show that sutured embedded contact homology is a natural invariant of
sutured contact 3-manifolds which can potentially detect some of the topology
of the space of contact structures on a 3-manifold with boundary. The appendix,
by C. H. Taubes, proves a compactness result for the completion of a sutured
contact 3-manifold in the context of Seiberg-Witten Floer homology, which
enables us to complete the proof of naturality.