George S. Ledyard
I was taught that you do give it back to them. But if you are going to lay it down, rather than hand it back, then you do so mindfully. Throwing it down is sloppy, is disrespectful of both the opponent and the audience. You never see anything like that in koryu.
Sensei Ledyard, I don't want to be disrespectful (because you are always a gentle person in your posts and a highly skilled Aikido Sensei), but I just want to point out that our dojo was taught this way by Paolo Corallini Shihan, 7th dan Aikikai (he has been promoted this week, congratulations!). Corallini Sensei spent great amounts of time with Saito Sensei who promoted him to 7th dan Takemusu Aiki (Iwama Ryu). Saito Sensei, as Corallini Sensei always tells in his seminars taught him this way also. Saito Sensei spent 23 years with O'Sensei. I don't know, but I have to guess, this is the way that O'Sensei used to train also, or at least, teach Saito Sensei.
I've heard the reasons for doing one way or another, and I sincerely, prefer doing it this way, so is the way our dojo expect us to. I understand that you do not see anything like that in koryu, but Aikido isn't a koryu, so I don't find that a valid argument.
Hope I explained myself clearly