By a paramedial groupoid we mean a groupoid satisfying the equation ax·yb=bx·ya. This equation is, in certain sense, symmetric to the equation of mediality xa·by=xb·ay and, in fact, the theories of both varieties of groupoids are parallel. The present paper, initiating the study of paramedial groupoids, is meant as a modest contribution to the enormously difficult task of describing algebraic properties of varieties determined by strong linear identities (and, especially,of the corresponding simple algebras).