High energy-density rechargeable batteries are strongly demanded from the viewpoint of energy and environmental concern. This work is devoted to fundamental electrochemistry on a novel concept of rechargeable batteries, "rocking-chair type" Mg-Li dual-salt batteries (DSBs), where both Mg and Li cations are carrier ions. In this system, dangerous dendritic growth is drastically suppressed by co-electrodeposition of Mg and Li, and Mg-Li alloys can be used as anode materials with high electrical capacities. As a DSB cathode material that can accommodate both Mg and Li cations, we use a spinel oxide MgCo2O4, in which an eccentric insertion mechanism, the "intercalation & push-out" process, occurs. Mg insertion occurs at 2.9 V vs. Mg2+/Mg and Li insertion occurs at 3.1 V vs. Li+/Li, being consistent with ab initio calculations, and its capacity approximately amounts to 150-200 mA h g-1. In the combination of MgCo2O4 and Mg50Li50 alloys, the cell voltage during discharge is as high as about 2-3 V. The concept of rocking-chair type DSB systems provides a new strategy for future safe rechargeable batteries combining high energy/power densities.