  The Chow rings of hyperK\"ahler varieties are conjectured to have a
particularly rich structure. In this paper, we focus on the locally complete
family of double EPW sextics and establish some properties of their Chow rings.
First we prove a Beauville-Voisin type theorem for zero-cycles on double EPW
sextics; precisely, we show that the codimension-4 part of the subring of the
Chow ring of a double EPW sextic generated by divisors, the Chern classes and
codimension-2 cycles invariant under the anti-symplectic covering involution
has rank one. Second, for double EPW sextics birational to the Hilbert square
of a K3 surface, we show that the action of the anti-symplectic involution on
the Chow group of zero-cycles commutes with the Fourier decomposition of
Shen-Vial.
