Strength comes from adversity. Adversity comes from struggle. Struggle comes from enemies. Enemies come from opposing natures. For every action there is an equal and opposite reaction. Every force, every idea, every object has its opposite.
Anything that is can be defined by its opposite: therefore, opposites give meaning. Without an opposite - without forces working against an object - the object atrophies and grows weak. Opposition brings conflict. From conflict, a body gains knowledge, power, and confidence. The weak are culled so that the strong can grow even stronger. When a cutter runs into the Opposition, it's best to assume the worst. More than likely, they're going to try to stop him from whatever he's doing. The only hope he's got is if the Opposers are already busy opposing someone or something else.
The many drawbacks to being a member of the Opposition are probably obvious. No one likes them, thinking them to be belligerent, contrary, and unfriendly. (All true.) Further, no Opposer can ever espouse any belief, opinion, motivation, or outlook other than that of the Opposition, except to oppose the belief of someone else. Black is white - unless someone says it's white, in which case it's black. The organization of this sect is particularly loose. Tiny contingents within the group keep it divided, enough so that there is virtually no chance of them ever becoming as large or as powerful as one of Sigil's factions.
Here's a tip for planewalkers who encounter an Opposer: Don't try the old "turn their belief on themselves trick" and tell an Opposer that if he really opposes everything, then he opposes himself and his own beliefs too. Every member of the Sect has heard that one, and the most common response is, "Now you're getting the idea, berk!" punctuated by a punch in the face. Their rationale is that, yes, they even oppose their own beliefs - or rather, they'd like others to oppose them, in order to make them stronger. Every belief must be challenged, constantly.
Each contingent of the Sect lives in one of the Inner Planes, with many favouring the Elemental Plane of Air due to their inability to dwell in most of the other planes long-term without a significant amount of magical aid. While all of the Inner Planes are equally important to the Opposition, occasionally one splinter group or another within the sect declares that one element is stronger than another, causing other groups to - surprise, surprise - oppose them. These sorts of activities are short-term and do more to encourage conflict than actually espouse a real preference for one plane. Opposers are rarely found on the Outer Planes. While they enjoy the clash of alignments and beliefs common to those planes, they prefer the more conspicuous manifestations of their beliefs in opposites embodied by the Inner Planes.
The Godsmen, who believe in the endless testing of everyone, understand the Opposers' love of adversity. Likewise, both the Fated and the Bleak Cabal agree in the idea that life is a struggle and that the strong should survive. However, the Opposition, due to the sect's very nature, has no allies. Nor does the sect want them. They oppose the beliefs of all other factions and sects (which makes them a little like the Anarchists, though the two groups still aren't allies). It's been said that if the Opposers were stronger or more significant, the other factions would bring them down on principal alone.