When solutes are small compared to the size of the ions in an ionic liquid, energetic heterogeneities associated with charge enhanced (stiff) and charge depleted (soft) nanoenvironments are sampled. In a recent article [J. C. Araque et al., J. Phys. Chem. B 119(23), 7015-7029 (2015)], we explored large deviations from Stokes-Einstein translational diffusion caused by such a heterogeneity. The current article is set to explore the effect of soft and stiff solvent environments (i.e., structure) on OH-bond rotations in the case of water and small alcohols in 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (Im1,2 (+)NTf2 (-)). Is solute rotational dynamics heterogeneous? If so, are solute rotations and translations coupled in the sense that stiff and soft solvent environments hinder or speed up both types of dynamics? For the systems studied here, there appears to be a clear connection between translations, rotations, and stiff/soft solvent environments. We also discuss interesting asymmetries of the correlation between solutes with anions and cations.