For those not super into the lore already:
FWC uses a fancy pants machine to see multiple possible futures and plans for them. They believe the only constant is war and we need to prepare for the fight, so they arm guardians and guide them to help influence the future we move towards.
New Monarchy are essentially your friends neighborhood fascists. Not hateful fascists, but people who believe the best way to ensure our survival is to unite everyone under one, unquestioned ruler to guarantee swift, uniform, decisive action.
Dead Orbit think if we stay on earth, in the city, near the traveler, we will die. They want to build a fleet and run away.
Dead Orbit are a bunch of cowards an quitters. They steal from Holliday, they want to abandon the Traveler to our enemies, and don't want to fight, but flee. They survive on their edgy name and edgy color scheme alone.
Dead Orbit is the worst faction. When the Cabal came, FWC fought back, New Monarchy took in civilians, and DO ran away.
No Guardian should willingly align with DO.
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I would only align with dead orbit if they've got good guns, or if their philosophy of exploring the solar system applied to the game and we got to see a bunch more moons and such in the solar system, maybe beyond. But since that doesn't happen and the only difference is colors and reskinned armor, I had to break my 3 year long tradition of running DO to play war cult, because I'm so bored with weapons in this game I wanted to try that semi-less boring FWC hand cannon