The most fundamental form of epistasis occurs between residues within a protein. Epistatic interactions can have significant consequences for evolutionary dynamics. For example, a substitution to a deleterious amino acid may be compensated for by replacements at other sites which increase its propensity (a function of its average fitness) over time - this is the evolutionary Stokes shift. We discovered that an opposite trend -the decrease in amino acid propensity with time- can also occur via the same epistatic dynamics. We define this novel and pervasive phenomenon as the evolutionary anti-Stokes shift. Our extensive simulations of three natural proteins show that evolutionary Stokes and anti-Stokes shifts occur with similar frequencies and magnitudes across the protein. This highlights that decreasing amino acid propensities, on their own, are not conclusive evidence of adaptive responses to a changing environment. We find that stabilizing substitutions are often permissive (i.e., expand potential evolutionary paths) whereas destabilizing substitutions are restrictive. We show how these dynamics explain the variations in amino acid propensities associated with both evolutionary shifts in propensities.