Any ideas how to decide which Lawful faction of the 3 is right for a char.
Heres what I have so Far.
Mercy Killers are the most Vengeful.
Guvners are Buerucratic.
Harmoniums members tend to be a bit more optimistic... but have a limited grip on reality.
Anything else to point out?
As stereotypes, the Harmonium are lawful good, the Guvners are lawful neutral, and the Mercykillers are lawful evil. Which is your character?
Of course you don't have to be evil to be a Mercykiller, and you certainly don't need to be good to be a Hardhead-- just ask the Indeps-- but it's a place to start.
Deconstructing that, as Planescape does, the Harmonium is a conquering army answerable to no one else, while the Mercykillers are a mercenary force that, nasty though they often are, are a tight-knit bunch that can at least be relied upon to largely police themselves. The Guvners are more akin to like-minded colleagues who happen to have an interest in common, a legal-academic-arcano-scientific community, by and large above reproach-- Guvners are trustworthy because they are Guvners; if a Guvner couldn't be trusted, then he wouldn't be a Guvner anymore.
So is your character looking for a cause to fight for, or a job or calling, or position, status, and maybe the answers to some nagging questions?
A Guvner searches for truth, for the natural order that underlies the multiverse and gives it structure, purpose and meaning. A Mercykiller desires justice, seeking out meaning after the fact, knowing that the world is inherently full of pain and misery, and ensuring that those who deal it out get their fair share. The Harmonium, meanwhile, aren't searching for anything. They've already found peace, and now they're out to spread it everywhere.
Truth, justice, peace. Three expressions of law. What's most important to your character?
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Guvners don't have to be bureaucratic, though that's how they're organized at present and you'd have to work within that system to rise in the faction. Even so, a Guvner character could plump for other forms of order while remaining inside the faction's established rules. Iconoclasm is a subjective thing.
On the other hand, I don't find the Harmonium to be particularly optimistic at all, and it seems like their grip on reality is all too concrete, and intimidating in its pragmatism. The idea that peace and good can only ever win out after a sufficient number of heads have been cracked seems like a very dark form of cynicism indeed.
And are justice and revenge ever the same thing? Most Mercykillers seem to think so, but maybe that's just the long-term Sodkiller influence talking.
The Harmonium know exactly what they're doing. Belief is power, power to shape the outer planes. If everyone was willing to sacrifice their precious individuality, surrender their egos, and stop arrogantly insisting on having beliefs of their own, everyone could believe the same thing. And with that level of power, they could do anything. They could turn the Lower Planes into heavens and eliminate war and strife.
To them, that's worth any cost.
Some Mercykillers are vengeful, but some just believe in justice, impersonal and without malice. If you break the law you must pay the price, whether you're a god, fiend, or mortal. It doesn't matter if the law is good or evil. It might even be okay to break the law as long as you pay afterward, and some might willingly do so. Still others believe justice must be balanced by mercy, with punishment suspended if the law can be preserved another way. If someone obviously isn't going to commit the crime again, why bother punishing them? Some Mercykillers think of themselves as living enforcers of karma, like the inevitables, while others are willing to grant clemency.
The Fraternity of Order is a group of scientists, explorers, bureaucrats, sages, and scholars. They don't want to create order because they believe the multiverse is already ordered. They just want to catalog, organize, and unveil the order that's already there.
Note that any faction except the Xaositects and probably the Anarchists can have lawful members, so a lawful character isn't restricted to those three.
Harmonium members may be optimistic in the sense that they have an organizational ideal to strive for. That doesn't necessarily mean that they have a limited grip on reality, any more than a faction that claims we can all become gods.
Your character may also want to decide between the three based on the roles they typically play in Sigil. Do you want to basically be a cop? That would be a traditional Harmonium role. A jailor, or bounty hunter? Many are Mercykillers. (Remember that Mercykillers may also include the Sons of Mercy, who believe in justice by the sword but not in sadism.)
If you have a particular character in mind, perhaps you could discuss with us what that character is interested in doing, or why they're interested in the lawful factions.