  Using the technique of Poincar\'{e} return maps, we disclose an intricate
order of the subsequent homoclinics near the primary homoclinic bifurcation of
the Shilnikov saddle-focus in systems with reflection symmetry. We also reveal
the admissible shapes of the corresponding bifurcation curves in a parameter
plane of such systems. The scalability ratio of geometry and organization is
proven to be universal for such homoclinic bifurcations of higher orders. Two
applications with similar dynamics due to the Shilnikov saddle-foci, a smooth
adaptation of the Chua circuit and a 3D normal form, are used to illustrate the
theory
