The solvation dynamics following the instantaneous creation of a positive or negative electronic charge in a previously neutral solute immersed in different water-dimethyl sulfoxide (DMSO) mixtures, spanning the entire composition range, is analyzed by molecular dynamics simulations. The solvation responses are strongly dependent on the sign of the solute charge, being considerably faster in the presence of cations for all mixtures considered. In terms of the composition dependence, the mixtures' solvation response to the creation of the anion departs substantially from the pure solvents', whereas for the cation, the mixtures' responses are close to those exhibited by pure DMSO. In the case of anions, the mixture overall solvation time, defined as the time integral of the nonequilibrium response, can be as large as ten times the solvation time in pure DMSO, the slowest of the two cosolvents. The DMSO contribution to the mixtures' solvation response may present an intriguing negative branch in the rotational-diffusion regime which persists for times much longer than the time scales typically found in other polar liquids and mixtures. This negative portion is nearly cancelled by an equally long-lasting positive contribution from water, resulting in a fast-decaying, total response curve which is typical of many polar liquid environments. This behavior is rationalized in terms of the time evolution of the first solvation shell around each type of solute.