Use a different package name.
com.example.version36.ThatThing
com.example.version37.ThatThing
This allows you to keep all the "things" for specific versions together.
Apache Commons Lang used this format when releasing their "version 3" rewrites.
Since you need to be able to reference both version in a single class, I would shorten the names to:
ThatThing36
ThatThing37
In your case, this seems to be descriptive enough, and it removes the ugly Impl and _'s.