  This article is about Lehn-Lehn-Sorger-van Straten eightfolds $Z$, and their
anti-symplectic involution $\iota$. When $Z$ is birational to the Hilbert
scheme of points on a K3 surface, we give an explicit formula for the action of
$\iota$ on the Chow group of $0$-cycles of $Z$. The formula is in agreement
with the Bloch-Beilinson conjectures, and has some non-trivial consequences for
the Chow ring of the quotient.
