  We extend non-emtpyness and irreducibility of Hassett divisors to the moduli
spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which
in turn provides a systematic approach for describing the irreducible
components of intersection of Hassett divisors. We show that Fermat cubic
fourfold is contained in every Hassett divisor, which yields a new proof of
Hassett's existence theorem of special cubic fourfolds. We obtain an algorithm
to determine the irreducible components of the intersection of any two Hassett
divisors and we give new examples of rational cubic fourfolds. Moreover, we
derive a numerical criterion for the algebraic cohomology of a cubic fourfold
having an associated K3 surface and answer a question of Laza by realizing
infinitely many rank $11$ lattices as the algebraic cohomologies of cubic
fourfolds having no associated K3 surfaces.
