There's only so much of this question I can answer without breaking the 20 year rule as the face of both these movements have changed significantly since 1998 but I will try my best.

Firstly I'd like to point out while Irish Nationalism/Republicanism is generally associated with the left wing and Unionism/Loyalism with the right wing it is not strictly true. The Alliance Party (while today a neutral centre party) was historically a left wing unionist party. Similarly PUP (a party with historic loyalist paramilitary links) were a unionist socialist party.

The Republican movement and push for independence in Dublin during the early 20th century was very much a socialist movement. James Connolly, one of the leaders of the 1916 Rising that ultimately lead to the formation of the IRA and Ireland's independence, was a well documented and committed Socialist, a strong supporter of Syndicalism and was aligned with the Industrial Workers of the World. The other leaders are also generally seen as Socialists and left-leaning but Connolly was by far the most vocal of it out of all the leaders. Irish independence was just as much a political uprising as it was a violent uprising. There was public outrage among the Irish public after the execution of the 1916 leaders and Sinn Féin (one of the main nationalist parties) used that as platform to build support for an independent Ireland eventually winning a majority in 1918. The Sinn Féin (and other Republican/Nationalist parties) of the 1970s and beyond would have looked towards the ideals of the socialists leaders of 1916 as an inspiration for the modern Irish Republican movement.

Edward Carson, widely considered both the father of Ulster Unionism and some would even say for the creation of Northern Ireland was a member of the Liberal Unionist Party which was aligned with England's Conservative Party and he eventually the leader of the Ulster Unionist Party, a centre right party. Similarly many (but not all) unionist leaders of later times would look towards Carson as an inspiration for their movement. I can't offer much more on why Unionism is traditionally right wing other than this without resorting to speculation.