We investigate how symmetry, exact coherent structures (ECSs), and their invariant manifolds organize spontaneous flow reversals in a 2D active nematic confined to a periodic channel. In minimal flow units commensurate with the intrinsic active vortex scale, we use equivariant bifurcation theory to trace the origin of dynamically relevant ECSs via a sequence of symmetry-constrained local and global bifurcations. At low activity level, we identify relative periodic orbits, created via a sequence of SNIPER, homoclinic and heteroclinic bifurcations, whose invariant manifolds provide robust heteroclinic pathways between left- and right-flowing nearly uniaxial states. These result in several symmetry-dictated reversal mechanisms in the preturbulent regime, with and without vortex-lattice intermediate states. In the active turbulent regime, this ECS skeleton persists and organizes chaotic attractors exhibiting persistent two-way reversals. By classifying ECSs through their symmetry signatures, we relate a small set of ECSs embedded in turbulence back to the preturbulent branches, and show that typical turbulent trajectories repeatedly shadow these ECSs and their unstable manifolds, resulting in near-heteroclinic transitions between opposite-flow states. Our results establish that channel confined active nematic turbulence is organized by a low-dimensional, symmetry-governed network of invariant solutions and their manifolds, and identify dynamical mechanisms that could be exploited to design, promote, or suppress flow reversals in active matter microfluidic devices.