The principal objective of the work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations are found in response to a number of questions in the survey article by R. Schneider and W. Weil [Convexity and its applications, Collect. Surv., 296-317 (1983; Zbl 0524.52002)]. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. A final set of characterizations is obtained in terms of extremal geometric inequalities.