My current ideas on the Confederation's probable political parties:
Traditional Proletarist: Workers' Commonwealth Party (Scandinavian proletarism), Labour Congress Party (Mathis-Douglass proletarism)
Social Proletarist: Socialist Party
Liberal: Septembrist Party, Republican Party
Moderate: --
Conservative: Party of Order, Conservative Party
Militarist: Party of the Confederate People
Religious: Catholic Party (moralism), Christian Democratic Party (protestant solidarity)
Nationalist/Separatist: --
For the immediate future, the defining features of the Septembrist Party vs. the Republican Party are that the Septembrists were the liberals who actually made unification happen, and wish to see the Confederation reincorporate other former Confederate states as well. Having had to do so much compromising during the Conference of Paris (so much goddamn compromising), they'll be far less than idealists, and will likely be pragmatically balancing various interest groups and lobbies, while trying to make keep their project of the Second Confederations united, and growing (probably through lobbying for national welfare services, and of course trying to get other countries to join).
Undoubtedly many people who support the Confederation are not Septembrists ideologically, but are still left-wing, and I'd therefore expect some kind of pro-republican-liberalism party to form, and thus, the Republican Party. Far more idealist, this party would not be quite as interested in expanding the Confederation, so much as adhering to its principles of prosperity, peace, democracy and so forth.
Then on the right. I would expect to basic sentiments from the more conservative side, first, the pragmatic "we're a Confederation now, so might as well run it the right way (the conservative way)", probably embodied in the Party of Order, which I understand under the First Confederation was a conservative party. The second kind of response would have to be one of disdain for reformation (or at least caution and unease), a party which has a country-wide solidarity over local autonomy. This kind of party would probably not advocate for the destruction of the Confederation (I'm sure many conservatives have some nostalgia or sentimental regard for the First Confederation), but would wish to see aristocracy in charge (due to no central monarchy) and would probably emphasis the autonomy of regions - not try and break the Confederation, though neither particularly solidify it. The name Conservative Party sufficiently sums up this in my mind.