It's a lot of things, but for me, it's a sense of emotional safety; that you can be fully vulnerable, open and honest about things with the person, and they will not judge you, criticize you, abandon you, etc.

If the push/pull stops does that mean you trust him?
Some folks do the push/pull to see if they can trust the clinician, some do because they know they can.
When a person does the push/pull, I think it's very possible that all the while they do trust the clinician. Maybe not consciously, but it's still possible to consciously trust them, and do the push/pull at the same time. This is because the client knows he/she can push and pull, and this person will still be here, still supportive (hopefully). And maybe the client needs to engage in this behavior to replay a trauma or fulfill some kind of desired role on either/both sides; there's an infinite number of possible needs behind it.
That said, push/pull could also mean of course that the person doesn't trust the clinician, maybe in the process of trying to get there. It just depends on many factors.