Snapping instabilities in slender structures offer an efficient pathway for rapid motion generation in soft robotics, enabling large geometric transitions without complex control systems. This work presents a novel tendon-driven mechanism based on spiral-geometry metabeams that exploits snap-through behavior to produce controllable reciprocating and non-reciprocating motion patterns. The developed snapping structure consists of interconnected Archimedean double-spiral unit cells arranged in series to form metabeams positioned at an angle. This spiral-based design strategy addresses a fundamental limitation in existing snapping mechanisms: the rate-dependent behavior and energy dissipation associated with soft polymers like thermoplastic polyurethane. By leveraging geometric compliance rather than material elasticity, the proposed design achieves large reversible deformations using polylactic acid, a relatively stiff thermoplastic material, while maintaining efficient elastic energy recovery with only 20\% energy dissipation. Finite element analysis and experimental characterization under different boundary conditions revealed that mechanical response, including critical forces and stability characteristics, can be precisely tuned through constraint design alone. The fixed-fixed configuration exhibited monostable behavior with a critical force of 2.74 N, while the pinned-pinned arrangement showed bistability at 1.24 N. Most notably, the asymmetric fixed-pinned configuration demonstrated distinct snap-back phenomena depending on loading direction, resulting in either reciprocating or non-reciprocating deformation trajectories. The practical utility of this mechanism was demonstrated through a proof-of-concept swimming robot with tendon-actuated fins. The non-reciprocating mode achieved forward propulsion of approximately 32 mm per 0.4 s cycle (81 mm/s), substantially outperforming the reciprocating mode (9.6 mm per cycle). These results highlight how geometric programming of snapping structures can generate efficient directional motion without sophisticated actuation systems, offering a promising approach for compact, energy-efficient soft robotic mechanisms.