We analyze how a set of 6 points of $\Rp 2$ in general position changes under quadratic Cremona transformations based at triples of points of the given six. As an application, we give an alternative approach to determining the deformation types (i.e. icosahedral, bipartite, tripartite and hexagonal) of 36 real Schlafli double sixes on any nonsingular real cubic surface performed by Segre.