The Restructurists are one of two factions of Rikti society, the other being Traditionalists. Most missions which involve Rikti who fight each other, defect to the Human side or otherwise seem to betray their collectivist ideals will involve members of both factions.
While the Traditionalists want to maintain the status quo of the Rikti empire, the Restructurists are not satisfied with the way things are. Peace is not good for business when you are a soldier and the Restructurist faction is run mainly by soldiers. They want to re-define Rikti politics to give a greater role to the military. The invasion of our dimension was engineered by the Restructurists, most likely to create a reason for the military to grow in size and power.
The Restructurists are also behind the transformation of Paragon City's homeless population into the Lost. This serves three purposes to them. First, it was the only way they could get reinforcements after Alpha Team sealed off their home dimension. Second, it gives the Rikti an opportunity to learn more about Human biology. Third, and most insidious, some missions texts and NPC dialogues hint that certain Restructurist medics were hoping that the Lost could be used as spare body parts for injured Rikti.
In missions in which you must rescue a Rikti or talk to one who is on your side, you are typically talking to a traditionalist. Most of what we know of the Restructurists comes from these Traditionalist NPC's.