Brachiation robots mimic the locomotion of bio-primates, including continuous brachiation and ricochetal brachiation. The hand-eye coordination involved in ricochetal brachiation is complex. Few studies have integrated both continuous and ricochetal brachiation within the same robot. This study seeks to fill this gap. The proposed design mimics the transverse movements of sports climbers holding onto horizontal wall ledges. We analyzed the cause-and-effect relationship among the phases of a single locomotion cycle. This led us to apply a parallel four-link posture constraint in model-based simulation. To facilitate smooth coordination and efficient energy accumulation, we derived the required phase switching conditions as well as joint motion trajectories. Based on a two-hand-release design, we propose a new style of transverse ricochetal brachiation. This design better exploits inertial energy storage for enhanced moving distance. Experiments demonstrate the effectiveness of the proposed design. A simple evaluation method based on the final robot posture from the previous locomotion cycle is applied to predict the success of subsequent locomotion cycles. This evaluation method serves as a valuable reference for future research.