Recently pseudomonotone variational inequalities have been studied quite extensively, hereby extending the theory of pseudoconvex minimization problems. The focus of the present work are "pseudoaffine maps", i.e., pseudomonotone maps T for which -T is also pseudomonotone. A particular case of such maps are the gradients of pseudolinear functions. Our main goal is to derive the general form of pseudoaffine maps which are defined on the whole space.