Ina recent paper the author proved a classification theorem for Atiyah-real vector bundles on spaces with free involutions. This result is now applied to the group of Atiyah-real line (i.e., one-dimensional) bundles on spheres, denoted Lr(S"). It is proved that such bundles are classified by maps into a complex quadric QCn. Using this classification it is proved that Lr(S1)=0 and that for «^3 the groups are all isomorphic.