Proving the irrationality of the generic complex cubic fourfold X is a major open problem. If X contains a plane, then there is an associated K3 surface S of degree 2 and a Brauer class on S, called the Clifford invariant of X. Hassett proved that if the Clifford invariant is trivial, then X is rational. Whether the converse holds was an open question. In this talk, I'll report on joint work with Marcello Bernardara, Michele Bolognesi, and Tony Várilly-Alvarado, constructing families of rational cubic fourfolds containing a plane with nontrivial Clifford invariant, thereby showing that the converse does not hold. Our approach uses classical Hodge theory as well as point counting techniques over finite fields. Finally, I will spell out the connection to Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.