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This study examines the structure of non-symmetric homoclinic trajectories and associated symmetric periodic-orbit families near the collinear equilibrium point L₃ in the planar photogravitational restricted three-body problem. The mathematical formulation incorporates radiation-reduction factors into the rotating-frame equations of motion, and fourth-order asymptotic expansions are developed to obtain initial approximations for the outgoing and incoming trajectories. The governing equations, equilibrium conditions and asymptotic solutions provide the mathematical framework for examining doubly asymptotic connections. Numerical integration and iterative phase-space matching are employed to investigate the orbital configurations, while MATLAB software is used to generate graphical representations of the asymptotic trajectories, zero-velocity curves and associated periodic-orbit families. The analysed configurations comprise five orbital classes, designated 1–2, 1–4, 1–6, 1–8 and 1–10, distinguished by their section-crossing patterns. The results illustrate variations in orbital geometry, equilibrium location and Jacobi constant across the considered mass and radiation parameters. The graphical representations further demonstrate the relationship between symmetry-related homoclinic branches and neighbouring symmetric periodic solutions, whose continued families approach limiting simple periodic orbits of the L₃ Lyapunov family. The study presents a mathematical formulation and graphical analysis of the orbital structures near L₃, providing a framework for investigating how photogravitational parameters influence homoclinic trajectories and periodic-orbit configurations.