We realize the half-steps of a general class of Markov chains as alternating projections with respect to the reverse Kullback-Leibler divergence between sets of joint probability distributions which admit an information geometrical property known as autoparallelism. We exhibit both a weak and strong duality between the Markov chains defined by the even and odd half-steps of the alternating projection scheme, which manifests as an equivalence of entropy decay statement regarding the chains, with a precise characterization of the entropy decrease at each half-step. We apply this duality to several Markov chains of interest, and obtain either new results or short, alternative proofs for mixing bounds on the dual chain, with the Swendsen-Wang dynamics serving as a key example. Additionally, we draw parallels between the half-steps of the relevant Markov chains and the Sinkhorn algorithm from the field of entropically regularized optimal transport, which are unified by the perspective of alternating projections with respect to an \alpha-divergence on the probability simplex.