<p>This study aims to provide comprehensive insights into the dynamics of retrograde co-orbital motion, contributing to our understanding of the origin and evolution of retrograde asteroids in the solar system. The paper studies retrograde co-orbital motion with Jupiter through the computation of symmetric periodic orbits in both planar and spatial circular restricted three-body problems. The research begins with an analysis of families of the planar circular problem at which there exist three main families of co-orbital motion. Starting from the vertical critical orbits of these families we extend our study to three-dimensional configurations, revealing that most spatial families exhibit instability. The phase-space structure is also explored by using dynamical stability maps under various initial conditions. These maps help identify regions of stable motion within predominantly chaotic domains. The theoretical framework is applied to asteroid (514107) Ka’epaoka’awela, demonstrating its chaotic nature with a Lyapunov time of approximately 6350 years, despite remaining in co-orbital resonance with Jupiter for many million years.</p>