Literally: join – vertical – lead – horizontal
Alternately: This compound commonly refers to diplomacy based on calculations of profit and loss; of making and breaking alliances based on the convenience of the moment. Originally, it referred to two competing schools of thought toward the end of the Warring States period in China. By around 334 BCE, the Qin state was becoming increasingly powerful compared to its neighbors, leading each of the other six states comprising “China” at the time to consider whether they should ally against the Qin (a “vertical” strategy), or ally with it in order to share in its success (a “horizontal” strategy).
(In the end, the Qin repeatedly exploited fears and tensions to lure its neighbors out of their “vertical” alliance into “horizontal” ones, allowing it to take over the other states and eventually gain complete dominion.)
Notes: More properly 従 means something like “follow” and 衡, something like “equilibrium.” In this case, though, they refer respectively to the vertical and horizontal axes on a map, and thus ultimately to alliances between northern and southern, or eastern and western, kingdoms.
Reading 合従 as goujuu, 連衡 as rengou, or writing renkou as 連合 are all considered errors.