Cyclic loading-unloading tests in tension and compression were carried out on pure Mg and alloys with 0.4, 1.5 and 4.2 at.% Gd, over a range of grain sizes, to quantify the solute and strain dependence of the materials anelasticity in the form of hysteresis loops. For a given grain size, the anelastic effect was more pronounced, i.e., the loops were wider, for pure Mg, and it decreased rapidly with the Gd concentration. The effect was larger for the finer grains in all materials, and in compression for the pure Mg and the 0.4Gd alloy. No difference between tension and compression was observed for the 1.5Gd alloy, whereas the loops were wider in tension than in compression for the 4.2Gd alloy. In comparison with existing studies in Mg-Al and Mg-Zn alloys, for a given solute concentration, Gd was more effective in reducing the magnitude of the effect than either Zn or Al, in that order. The overall behavior is discussed in terms of the hardening effects of short range order of the alloys on {1012} and {1011} twinning.