I would argue that there are a few ways this phrase can be inverted:
No rights reserved
Implies that the author waives all rights to his/her work (i.e. ultimately places it in the public domain)
All rights included
I've seen this one in the context of royalty-free music being used in the commercial sense, where if you pay for the license, you can use that song anytime anywhere, with all rights to the song. This seems to be the opposite of "All rights reserved" which we should know by now what it means
Copyleft
While not really a phrase, it is the opposite of copyright itself. Used primarily in software but maybe some other media can also be considered copyleft. As far as I'm aware, it has some ties with copyright itself (that you cannot remove attribution from the author, and, in case of software, must distribute it as is, without putting any restrictions yourself)
There are probably more means other than what I've listed, and I would love to hear your thoughts and suggestions!
I had the same question dozen years ago when was taking part in publishing one book (in my translation from French). Publishing house finally used phrase which in English should sound like: "Not protected by copyright law".