Initially there were two unions both founded in 1874. The Irish Football Union had jurisdiction over Clubs in Leinster, Munster and parts of Ulster; the Northern Football Union of Ireland controlled the Belfast area. The IRFU was formed in 1879 as an amalgamation of the two different organisations and branches were formed in Leinster, Munster and Ulster. The Connacht Branch was formed in 1886.(Wikipedia)

The IRFU predates partition... it was the soccer crowd that split:

Upon the partition of Ireland in 1921 the FAIFS (now the FAI) was set up to regulate the game in the Irish Free State (now the Republic of Ireland). Those behind the FAIFS believed that soccer should be regulated by a federation based in the Free State capital Dublin. The IFA's supporters argued that the federation should be based where the game was mainly played - Ulster and its principal city Belfast. Both federations claimed to represent the whole of the island and both competed as Ireland and both picked players from the two rival leagues - which also split at this time.
(Wikipedia)

Which endorses Helen's view I guess: the upper and middle class sport stayed as one because it would have been dominated by rich Protestant or Anglo-Irish and the working class sport split over politics.

While I didn't go to a "rugby school", my father played  at some level in London in the sixties...