It is conjectured that Jordan pairs are strongly related to a certain class of point-line spaces. These point-line spaces are related to twin buildings whose point-line truncations are parapolar. We will give a characterisation that makes use of convex spans of two points at finite distance and of an opposite relation on the set of points. It turns out that all these point-line spaces are exactly the point-line truncations of 6 well-known classes of geometries: Grassmannians of projective spaces, polar spaces, dual polar spaces, half-spin geometries, E6- and E7-geometries. Except for the latter two, the ranks of these geometries are not necessarily finite.