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!
The Creative Commons Zero / Public Domain Dedication / CC0 is the closest thing to a legally enforceable antonym of copyright.
and for those that don't like CC licenses applied to code, 0BSD is also an option
CC0 is the one CC licence you can safely use for code, as per the official recommendations. For all other CC licences, it is (strongly) discouraged.
What about the Unlicense?
Weird/confusing name, questionable legality and the website went down a while back (while mentioned explicitly in the licence...)
Use CC0 1.0 or Zero Clause BSD instead. They are more reputable, and all decent "public domain equivalent" licences are... well, equivalent in effect, anyway.
Unlicense only works in some legal systems. You cannot put your own work into the public domain in many european nations for example.
See this link: https://softwareengineering.stackexchange.com/a/147120
afaik some people are worried about the "legal enforceability" of the unlicense, which is funny given the point of it is to be an explicit "go do what you want" license.
if it explicitly granted patent rights i would definitely use it for software tbh