I tried and it didn't allow me to register a 10.3 Web Adaptor with a 10.2.2 ArcGIS Server Site.
From the server installation guide found here: Installing ArcGIS Web Adaptor—Installation Guides | ArcGIS for Server
"The ArcGIS Web Adaptor version must always match the version of its registered ArcGIS Server site."
So the way I read this, if you have 10.2 servers, you will need to install 10.2 web adaptors
Thanks for the responses guys. You saved me trying it this weekend.
Having now migrated to a new Webserver I wanted to update you on what I found. I decided to try it anyhow.
I have successfully registered 10.2.1, 10.2.2 and 10.3 ArcGIS Servers with 10.3 web adapters. It was completely painless and has worked a treat.
It appears that 10.3 web adapters are backward compatible. Well this is my experience anyhow.