  We study a correspondence between double EPW cubes and double EPW sextics,
two families of polarized hyper-K\"ahler manifolds related to Gushel--Mukai
fourfolds. We infer relations between these families in terms of Hodge
structures and moduli spaces of elliptic curves.
  As an application, we prove that a very general double EPW cube is the moduli
space of stable objects with respect to a suitable stability condition on the
Kuznetsov component of its corresponding Gushel--Mukai fourfolds; this answers
a problem posed by Perry, Pertusi and Zhao.
