A look around the net confirms that mainstream and liberal religion is instinctively left-wing, traditional and orthodox religion right-wing. The exceptions usually seem a little artificial, more a matter of sticking with a theory than immediate unselfconscious perception of how things are.
I suppose the reason things sort out that way is that both mainstream liberal religion and the Left accept the ideal of comprehensive control and reconstruction of human experience for human purposes while orthodoxy and the Right don’t. I expect orthodoxy to be the more durable outlook. Apart from the question whether we’ll really be able to control our world enough to ignore the things that are beyond our control, it’s not clear why the Left needs religion. If the idea is that man is going to create a fully human world why not just do it?