  Couplings play a central role in the analysis of Markov chain convergence and
in the construction of novel Markov chain Monte Carlo estimators, diagnostics,
and variance reduction techniques. The set of possible couplings is often
intractable, frustrating the search for tight bounds and efficient estimators.
To address this challenge for algorithms in the Metropolis-Hastings (MH)
family, we establish a simple characterization of the set of MH transition
kernel couplings. We then extend this result to describe the set of maximal
couplings of the MH kernel, resolving an open question of O'Leary et al.. Our
results represent an advance in understanding the MH transition kernel and a
step forward for coupling this popular class of algorithms.
