We reveal the existence of a new codimension-1 curve that involves a topological change in the structure of the chaotic invariant sets (attractors and saddles) in generic three-dimensional dissipative systems with Shilnikov saddle foci. This curve is related to the spiral-like structures of periodicity hubs that appear in the biparameter phase plane. We show how this curve configures the spiral structure (via the doubly superstable points) originated by the existence of Shilnikov homoclinics and how it separates two regions with different kinds of chaotic attractors or chaotic saddles. Inside each region, the topological structure is the same for both chaotic attractors and saddles.