The rate of evolution differs between protein sites and changes with time. However, the link between these two phenomena remains poorly understood. Here, we design a phylogenetic approach for distinguishing pairs of amino acid sites that evolve concordantly, i.e., such that substitutions at one site trigger subsequent substitutions at the other; and also pairs of sites that evolve discordantly, so that substitutions at one site impede subsequent substitutions at the other. We distinguish groups of amino acid sites that undergo coordinated evolution and evolve discordantly from other such groups. In mitochondrion-encoded proteins of metazoans and fungi, we show that concordantly evolving sites are clustered in protein structures. By analysing the phylogenetic patterns of substitutions at concordantly and discordantly evolving site pairs, we find that concordant evolution has two distinct causes: epistatic interactions between amino acid substitutions and episodes of selection independently affecting substitutions at different sites. The rate of substitutions at concordantly evolving groups of protein sites changes in the course of evolution, indicating episodes of selection limited to some of the lineages. The phylogenetic positions of these changes are consistent between proteins, suggesting common selective forces underlying them.