Cult of Sirius
Official Entry: " This bizarre cult has whipped up a religious frenzy following the appearance of the Dimensional Gates. The cult has long believed in redemption of the human race by a superior Alien race. They believe the UFOs and Aliens to be harmless and are rapidly gaining credibility and recruits from the general populations. This represents a considerable threat to X-COM because the cultists will do anything to assist the Aliens in their purpose, whatever that might be."
The Cult of Sirius do not provide any services to X-COM. At the beginning of the game they will issue absurd demands for "compensation" for X-COM's anti-Alien activities, but they will give up on this fairly quickly. After that, the only time X-COM will encounter the Cult is if either organisation raids the other's buildings.
Note that despite their hostility to X-COM, the Cult of Sirius is overall a fairly peaceful movement and will only rarely attack X-COM bases. However, they will become far more aggressive if infiltrated by the aliens, and the Cult can be infiltrated very easily (they will not report aliens in their temples, and infiltration happens twice as fast as normal).