Couplings play a central role in the analysis of Markov chain convergence and in the construction of new Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. Tight bounds and efficient methods require an appropriate choice of coupling, which can be hard to find when the set of possible couplings is intractable. To address this challenge for the Metropolis--Hastings (MH) family of algorithms, we establish a simple characterization of the set of all 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. [2021]. We conclude with a series of examples to build intuition. Our results represent an advance in understanding the MH kernel and a step forward for coupling this popular class of algorithms.
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