  A locally complete family of projective symplectic 4-folds is provided by
natural double covers of Eisenbud-Popescu-Walter (EPW) sextic hypersurfaces in
projective 5-space. The dual of an EPW sextic is another EPW sextic, thus to a
double EPW sextic X we may associate a "dual" double EPW sextic X^{\vee}. We
show how to obtain the Hodge structure on H^2(X^{\vee}) from that on H^2(X); it
turns out that they are isomorphic over the rationals but not over the integers
(in general).
