Hi Matthew, I haven't read the whole thing but this thread
talks about this.
I think I agree that the Iwama usage has imparted a nuanced meaning to me regarding this word. (The thread brings this up and points out that in Iwama namings, the "awase" practices often involve both partners doing the same movements.) So in the end the word seems to mean that there is some kind of "spatial matching" in addition to the "timing matching" Peter mentioned.
Basically, 2 things fitting together like puzzle pieces, filling each others' openings.
I'm always willing to refine my understandings of nebulous words though, if I get them wrong.