I imagine so! This has (to my knowledge) never been explained in great detail, but personally I've always interpreted the Ordo-Chamber relationship in a way that it basically "cuts off" the relevant organisation from its ordinary obligations towards the Inquisition. In other words, a Deathwatch Marine approached by an Inquisitor not belonging to the Ordo Xenos is under no obligation to obey their commands - even though the Inquisitor's authority is (officially) still absolute! I suppose you could say the Marines' vows apply only to the Ordo Xenos, essentially "normalising" their relationship to other members of the Inquisition to being similar to a normal Space Marine Chapter, thus allowing them to decide whether or not to help out.
It's the same with the Grey Knights and the Sisters of Battle. In a way, it's like the Inquisitors of one Ordo banding together and "calling dibs" on a military faction, which then might consider itself at liberty to refuse requests by other Inquisitors by pointing at the Ordo affiliated with them, saying they're too busy and that the supplicant should please talk with Inquisitor soandso.
In reality, of course it'd be somewhat more complicated in that the Ordos themselves are only semi-official constructs - "circles" of likeminded Inquisitors who are sharing information and resources. Inquisitor who are particularly powerful and so influential that their reach transcends the borders of their faction might still be reckoned with, and lesser Inquisitors might shy away from moving against claims made towards their Chamber Militant for the sake of keeping the peace.
I'd expect this to be exceptions from the normal dealings, however.
I'm actually curious how others think this works, now, so anyone feel free to chime in! Lynata said this.
The fluff says that Deathwatch 'belongs' to the Ordo Xenos, Grey Knights to Ordo Malleus and Sisters of Battle to Ordo Hereticus. I've always imagined it with Space Marines the combat minded but at the same time tactically genius Inquisitors get the more influence cause Space Marines don't care for money or power. They respect people who do things not the pen pushers of the Administratum. That's what makes (can't remember the name) Inquisitor of the Chamber where she is. She can talk like a diplomat and willingly fights and does the dirty work. So the Marines know not to mess with her.
It's not the fact that you have say lots of influence from the High Lords of Terra themselves, no it is from your self. What you do and choices you make, thats what Space Marines respect.
You seem confident that the Angels of Death will listen to you. Why is that? Ho Ho Ho! Because you have authority from the Lords of Terra. Lad thats not what they're looking for. I have served Deathwatch for years. You mean I mean with Deathwatch? I know what I mean! Look here, you have to listen and learn, that's the way. Oh, do answer me a question. How many scars have you earned? How many battles have you fought in? None? Forget it lad. They won't listen to a pen pusher like you. Wait, you say that you have about three years to deliver the message? Good! Give me the message and tell you what I'll even help you. Why don't you take a little holiday to a planet I know. Nice planet with lush jungles and friendly creatures, oh the planet is called Ravacene but don't worry nothings going to hurt you...
Spoken by Inquisitor Ol'Jackson to Inquisitor Marius Helburg.
Edited by Misha, 05 July 2014 - 05:47 AM.