The most basic meaning of awase for me comes from bukiwaza and means to match the rytm of the attacker in way that the attacker can´t see/feel my movement. In the basic exercises I have been told that we are supposed to match the breath of uchitachi.
Very interesting! Thank you, Peter! I can see how what I've practiced could fit with that description as well. I need to get to the dojo more to be sure though.