  A maximal coupling of two diffusion processes makes two diffusion particles
meet as early as possible. We study the uniqueness of maximal couplings under a
sort of "reflection structure" which ensures the existence of such couplings.
In this framework, the uniqueness in the class of Markovian couplings holds for
the Brownian motion on a Riemannian manifold whereas it fails in more singular
cases. We also prove that a Kendall-Cranston coupling is maximal under the
reflection structure.
