It has been stated that Elias James Corey, also a Nobel Prize winner, feels he is responsible for the ideas that laid the foundation for this research, and that Woodward unfairly neglected to credit him in the discovery. In a 2004 memoir published in the Journal of Organic Chemistry, Corey makes his claim to priority of the idea: "On May 4, 1964, I suggested to my colleague R. B. Woodward a simple explanation involving the symmetry of the perturbed (HOMO) molecular orbitals for the stereoselective cyclobutene to 1,3-butadiene and 1,3,5-hexatriene to cyclohexadiene conversions that provided the basis for the further development of these ideas into what became known as the Woodward–Hoffmann rules".
Corey, then 35, was working into the evening on Monday, May 4, as he and the other driven chemists often did. At about 8:30 p.m., he dropped by Woodward's office, and Woodward posed a question about how to predict the type of ring a chain of atoms would form. After some discussion, Corey proposed that the configuration of electrons governed the course of the reaction. Woodward insisted the solution would not work, but Corey left drawings in the office, sure that he was on to something.
"I felt that this was going to be a really interesting development and was looking forward to some sort of joint undertaking," he wrote. But the next day, Woodward flew into Coreys office as he and a colleague were leaving for lunch and presented Coreys idea as his own – and then left. Corey was stunned.
In a 2004 rebuttal published in the Angewandte Chemie, Roald Hoffmann denied the claim: he quotes Woodward from a lecture given in 1966 saying: "I REMEMBER very clearly—and it still surprises me somewhat—that the crucial flash of enlightenment came to me in algebraic, rather than in pictorial or geometric form. Out of the blue, it occurred to me that the coefficients of the terminal terms in the mathematical expression representing the highest occupied molecular orbital of butadiene were of opposite sign, while those of the corresponding expression for hexatriene possessed the same sign. From here it was but a short step to the geometric, and more obviously chemically relevant, view that in the internal cyclisation of a diene, the top face of one terminal atom should attack the bottom face of the other, while in the triene case, the formation of a new bond should involve the top (or pari passu, the bottom) faces of both terminal atoms."
In addition, Hoffmann points out that in two publications from 1963 and 1965, Corey described a total synthesis of the compound dihydrocostunolide.  Although they describe an electrocyclic reaction, Corey has nothing to offer with respect to explaining the stereospecificity of the synthesis.
This photochemical reaction involving 6 = 4×1 + 2 electrons is now recognized as conrotatory.