  There are three types of involutions on a cubic fourfold; two of
anti-symplectic type, and one symplectic. Here we show that cubics with
involutions exhibit the full range of behaviour in relation to rationality
conjectures. Namely, we show a general cubic fourfold with symplectic
involution has no associated K3 surface and is conjecturely irrational. In
contrast, we show a cubic fourfold with a particular anti-symplectic involution
has an associated K3, and is in fact rational. We show such a cubic is
contained in the intersection of all non-empty Hassett divisors; we call such a
cubic Hassett maximal. We study the algebraic and transcendental lattices for
cubics with an involution both lattice theoretically and geometrically.
