The term anti-invariant seems to be in use for what I believe is this general idea.
The Wikipedia glossary of invariant theory defines an anti-invariant as
A relative invariant transforming according to a character of order 2 of a group such as the symmetric group.
which, if my understanding of the relevant terms is correct, is a special case of what I want.
an "anti-invariant Riemannian submersion" involves a function not mapping a set to itself, but instead mapping a set to its orthogonal complement (defined e.g. here).
Other uses of the term "anti-invariant" can be found by google, e.g. here and here, though it's hard to jump in without context and figure out what the term means.
In conclusion, I think that if one were in great need of a concise term to use repeatedly, anti-invariant would be a good choice. But for most purposes it is probably best to write out "$Px \implies \lnot P f(x)$" explicitly, or in words.