When you figure what the assumption is, it's basically guaranteed to make you agree with each other: "Ah, but so you believe A! Of course you'd argue for A' then, makes total sense now!" It's then much easier to find compromises since you can work on the small thing below rather than the whole scaffold built on top.
Unfortunately I haven't yet found a way to easily discover what the differing assumption is aside from lengthy debates. This makes finding compromises with people harder as you can only discuss in depth with people where you can trust that they are truly arguing in good faith.