You could not, for example, be a member of the Godsmen and the Athar at the same time, or maybe you could as long as you don't want to become the kind of diety the Athar believe are false (they believe true deities could exist, but they aren't the kind we know of that many think are deities)
Do factions perhaps take the same philosophy towards membership as a monogamous marriage? That your devotion must be 100% towards the faction and no other? So, even though their goals might not contradict they consider it a fake commitment if you are in more than one at the same time?
2. Factions would certainly not like their members to have conflicted loyalties. Even factions who are more or less allied would want to make sure that, in a pinch, they could count on their members to defend them and not have to worry about their members suddenly picking the other side. So, I don't know what you'd have to do to get your name on the books in two factions and get away with it.
Faking it, of course, either with a pocket full of faction badges or being an Anarchist spy, is ironically straightforward in this regard.
3. Legitimately believing in two faction philosophies? Gaining powers through your belief in multiple factions? I don't know about this. I don't think I'd agree with the concept of someone being a full-out factioneer in mutliple factions. It just seems like warping reality, even in the mutable reality of Planescape, is hard. Too hard to do unless you're fully committed to that belief. But it's too easy to let someone get away with a character whose belief is "Faction A + Faction B" anyway. Make it something more interesting.
What about a character who believes that the multiverse is a prison and has to be torn down to escape from it? You could be some kind of combination of Doomguard and Dustman. Or someone who feels that the Cadence of the Planes is a force of universal equality and balance, and that it is rendered out of balance by injustice? If you can only gain your trance-like clarity when in pursuit of justice, you might be a Cipher/Mercykiller combination. Or, even weirder, a Harmonium-Revolutionary League split. You think that everyone's corrupt except the Harmonium, whose core ethos of everyone working together is the only chance to save it all. You want to bring down everything except for what you see as humanity's chance at salvation, but if either faction found out what you were really up to, they'd almost certainly kill you.
But it's not just putting two factions on your character sheet. I'd say you pick one that you officially belong to, and you have a somewhat-rogue set of beliefs. Maybe it's near-heretical for your faction, and your superiors don't approve, or you even keep it a secret from them. But in terms of the powers that a character could gain, I think the DM should be involved in crafting some kind of compromise.
Getting the best of both worlds is probably too much. If you use 2e, you'll have to make it up. If you use planewalker's 3e faction feats, you could get access to both factions' sets, or a limited selection, or get access to one faction's set with the right to choose DM-approved feats from the other. That part's not my job.
You just have to have one of those Factions be the Revolutionary League.
