As I currently understand the concept, awase (matching) means that neither uke nor tori has sente (initiative); both share the center until one of them takes it.
We've been practising the first six or seven partner practices with bokken from Saito sensei, and so far, beginners are struggling to match each other and being late, but with some of the more experienced people I've had the experience of moving completely together, as if their sword draws mine up or vice versa. I've been thinking that it feels better, but it feels like there should be a step further in the interaction where I can somehow take the initiative more. Not there yet obviously but this discussion seems to confirm that suspicion.
In empty hand it's not so obvious, so far I can only feel this with the sword work...