Rate variation among the sites of a molecular sequence is commonly found in applications of phylogenetic inference. Several approaches exist to account for this feature but they do not usually enable the investigator to pinpoint the sites that evolve under one or another rate of evolution in a straightforward manner. The focus is on Bayesian phylogenetic mixture models, augmented with allocation variables, as tools for site classification and quantification of classification uncertainty. The method does not rely on prior knowledge of site membership to classes or even the number of classes. Furthermore, it does not require correlated sites to be next to one another in the sequence alignment, unlike some phylogenetic hidden Markov or change-point models. In the approach presented, model selection on the number and type of mixture components is conducted ahead of both model estimation and site classification; the steppingstone sampler (SS) is used to select amongst competing mixture models. Example applications of simulated data and mitochondrial DNA of primates illustrate site classification via 'augmented' Bayesian phylogenetic mixtures. In both examples, all mixtures outperform commonly-used models of among-site rate variation and models that do not account for rate heterogeneity. The examples further demonstrate how site classification is readily available from the analysis output. The method is directly relevant to the choice of partitions in Bayesian phylogenetics, and its application may lead to the discovery of structure not otherwise recognised in a molecular sequence alignment. Computational aspects of Bayesian phylogenetic model estimation are discussed, including the use of simple Markov chain Monte Carlo (MCMC) moves that mix efficiently without tempering the chains.