Every reversible Markov chain defines an operator whose spectrum encodes the convergence properties of the chain. When the state space is finite, the spectrum is just the set of eigenvalues of the corresponding Markov transition matrix. However, when the state space is infinite, the spectrum may be uncountable, and is nearly always impossible to calculate. In most applications of the data augmentation (DA) algorithm, the state space of the DA Markov chain is infinite. However, we show that, under regularity conditions that include the finiteness of the augmented space, the operators defined by the DA chain and Hobert and Marchev's (2008) alternative chain are both compact, and the corresponding spectra are both finite subsets of [0,1). Moreover, we prove that the spectrum of Hobert and Marchev's (2008) chain dominates the spectrum of the DA chain in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. As a concrete example, we study a widely used DA algorithm for the exploration of posterior densities associated with Bayesian mixture models (Diebolt and Robert, 1994). In particular, we compare this mixture DA algorithm with an alternative algorithm proposed by Fruhwirth-Schnatter (2001) that is based on random label switching.