While looking at a representation theory question, I came up with the following sort of object. I want to know if it comes up often in combinatorics or some other area of mathematics.

Let $P$ be a finite poset and $a:P\to P$ be an order-reversing bijection. Then the lattice $J(P)$ of order ideals in $P$ gets an order reversing bijection by $I\mapsto (P-I^a)$, where $I^a$ denotes the image of $I$ under ${a}$.

For example, if $P$ is an antichain and $a$ is the identity, then this is a complement operation on the lattice. But in contrast, if $P$ is a chain, then you do not get a complemented lattice.