  We prove that the vast majority of JC*-triples satisfy the condition of
universal reversibility. Our characterisation is that a JC*-triple is
universally reversible if and only if it has no triple homomorphisms onto
Hilbert spaces of dimension greater than two nor onto spin factors of dimension
greater than four. We establish corresponding characterisations in the cases of
JW*-triples and of TROs (regarded as JC*-triples). We show that the distinct
natural operator space structures on a universally reversible JC*-triple E are
in bijective correspondence with a distinguished class of ideals in its
universal TRO, identify the Shilov boundaries of these operator spaces and
prove that E has a unique natural operator space structure precisely when E
contains no ideal isometric to a nonabelian TRO. We deduce some decomposition
and completely contractive properties of triple homomorphisms on TROs.
