Duality has always played a pivotal role in western culture[sic] and in philosophy at large. It also has its place in planescape.
I've been thinking about it and here is some of what I've come up with. Some of the factions are actually joined together by being on two sides of a duality.
The most obvios are the Fated and Ring Givers. As one lost power the other's power increased.
Another less obvios one is the Fraternity of Order and the Xaositecs. Their duality is around the nature of the universe and whether it is one of chaos or law. The Guvners seek to discover the laws governing the multiverse while the Xaositecs seek to discover the chaos that governs it. In the end they are really looking for the same thing.
Of coarse this isn't to say that ALL the factions fit neatly into dualities. Trinities work as well in the setting such as possably the Harmonium (universe ends in a great order), the Dustmen (The universe ends when its affairs are finished, aka the true death), and the Doomguard (universe ends in entropy).
The Fated and Ring-Givers sometimes have quite a lot in common. Ring-Givers often give only so that they themselves will benefit; their methods differ from the Fated, but not their motivation. Of course, other Ring-Givers really are as generous as they seem, but the Fated may be generous too, in a "teach a man to fish" sense. They can help people as much as they want; like the Ring-Givers, they expect to be paid back one way or another.
A better opposition is between the Fated and the Communals. They're laisse-faire capitalism and utopian socialism.
Another opposition might be the Opposition sect and the Sign of One. The Signers believe everyone is part of the same godlike soul; the Opposition believes in dichotomies, pairs, opposites. They're more like a Sign of Two.
The Harmonium and the Transcendent Order are often contrasted, as both seek harmony of very different kinds.
The Dustmen and the Believers of the Source are a good opposition. Both believe in reincarnation and a form of "enlightenment" at the end; the Dead believe this enlightenment takes the form of nothingness, while the Godsmen believe it is apotheosis.
The Sons of Mercy and the Sodkillers are another opposing pair, which makes the fact that they were once the same faction all the more ironic.
The Mercykillers and the Dispossessed sect might be another opposition; the jailers and the jailed.