This study examines the conditions governing the acceleration and reacceleration of cosmic rays (CRs) by ultra-relativistic jets. We demonstrate that cylindrically symmetric jets cannot reaccelerate external CRs. Without symmetry breaking, e.g., significant CR scattering, the jet-boosted magnetic and motional electric fields lead to specular reflection of incoming CRs. The one-dimensional particle dynamics, resulting from jet symmetry, categorizes CRs into passing particles and those bound by the jet's magnetic and motional electric fields. Passing particles exhibit energy consistency, originating and returning to the jet exterior with the same energy, necessitating trapping through scattering or other orbit distortions to access the jet's energy reservoir. These findings challenge some claims in the literature suggesting that outer CRs can achieve up to $2\Gamma^{2}$ energy gain in a "one-shot" scenario, briefly encountering a jet with a bulk Lorentz factor $\Gamma$ and experiencing no significant scattering within it. We establish that scattering within the jet facilitates CRs to traverse the separatrix between passing and trapped particles, thus potentially unlocking the jet energy for CRs. Additionally, we investigate the magnetic pumping acceleration mechanism, which requires CR trapping unless they are initially seeded within the jet. We illustrate how CR acceleration via magnetic pumping is contingent upon CR scattering. Consequently, our results underscore the critical role of particle scattering and symmetry-breaking mechanisms in efficiently accelerating particles through conducting fluid motions.