Which ergodic averages have finite asymptotic variance?

We show that the class of L2 functions for which ergodic averages of a reversible Markov chain have finite asymptotic variance is determined by the class of L2 functions for which ergodic averages of its associated jump chain have finite asymptotic variance. This allows us to characterize completely which ergodic averages have finite asymptotic variance when the Markov chain is an independence sampler. From a practical perspective, the most important result identifies a simple sufficient condition for all ergodic averages of L2 functions of the primary variable in a pseudo-marginal Markov chain to have finite asymptotic variance.