Is there any selfdual lattice $(X,\le)$ such that there is not any selfduality $f:X\to X$ such that $f\circ f = 1_X$?
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Yes. Let $L$ be the lattice structure on $\mathbb Z$ with the following Hasse diagram:
where all the diagonal arrows go upwards. It is easy to see that the only selfdualities of $L$ are of the form $f(n)=n+c$ for $c\equiv2\pmod4$, and in particular, they are never involutive. 

