There is unfortunately a lot of misinformation about medieval Irish kingship out there, so I'll start by saying that the position of "High King" did not rotate between four clans and was in fact a fairly late invention. Throughout the Irish annals, there are occasional references to someone who held the position of high king, usually "with opposition," meaning that he had to fight off other leaders who disputed his claim. Ireland had no centralized government until the arrival of the Anglo-Normans, and even then it took centuries to impose any sort of unified government over the island. Instead, Ireland was divided into many regions who were ruled by local leaders. There were "tribes" (although that word has misleading connotations for a lot of westerners), but family associations were as often rivalries as alliances. These local kingdoms, or tuaths, were connected to one another through overking-underking relationships. Underkings owed loyalty and tributes of cows to overkings. The position of high king was sometimes accorded to kings who were able to assert their dominance over large portions of the island. The Ui Neill dominated the political landscape for much of the eighth and ninth centuries, and they are the ones who established a lot of the language and tradition we associate with the position of high king, including the coronation at Tara. The idea of the high king culminated with Brian Boru, the last man to hold that sort of widespread power until the arrival of the Anglo-Normans, but he died defending his title in 1014. You might have seen references to the reenactment this past week of the Battle of Clontarf on its millennial anniversary.

All this is to say that yes, Irish clans did war amongst themselves similarly to Scottish clans. There has been substantial debate over the nature of that warring and whether violence is really a good way to characterize it, but there was certainly not a unified Ireland in the Middle Ages.

Sources: Ireland in the Middle Ages, Sean Duffy A New History of Ireland, vol. II (ed. Art Cosgrove), esp. "The Trembling Sod" by F.J. Byrne Irish Kings and High-Kings, F.J. Byrne Ireland Before the Normans, Donncha Ó Corrain From Kings to Warlords, Katharine Simms