Woodward and Hoffmann developed the pericyclic selection rules after performing extensive orbital-overlap calculations.  At the time, Woodward wanted to know whether certain electrocyclic reactions might help synthesize vitamin B.  Chemists knew that such reactions exhibited striking stereospecificity, but could not predict which stereoisomer a reaction might select.  In 1965, Woodward–Hoffmann realized that a simple set of rules explained the observed stereospecificity at the ends of open-chain conjugated polyenes when heated or irradiated.  In their original publication, they summarized the experimental evidence and molecular orbital analysis as follows:
*In an open-chain system containing 4n π electrons, the orbital symmetry of the highest occupied molecule orbital is such that a bonding interaction between the ends must involve overlap between orbital envelopes on opposite faces of the system and this can only be achieved in a conrotatory process.
*In open systems containing (4n + 2) π electrons, terminal bonding interaction within ground-state molecules requires overlap of orbital envelopes on the same face of the system, attainable only by disrotatory displacements.
*In a photochemical reaction an electron in the HOMO of the reactant is promoted to an excited state leading to a reversal of terminal symmetry relationships and stereospecificity.
In 1969, they would use correlation diagrams to state a generalized pericyclic selection rule equivalent to that now attached to their name: a pericyclic reaction is allowed if the sum of the number of suprafacial 4q + 2 components and number of antarafacial 4r components is odd. .  
In the intervening four years, Howard Zimmerman and Michael J. S. Dewar proposed an equally general conceptual framework: the Möbius-Hückel concept, or aromatic transition state theory.   In the Dewar-Zimmerman approach the orbital overlap topology (Hückel or Möbius) and electron count (4n + 2 or 4n) results in either an aromatic or antiaromatic transition state. 
Meanwhile, Kenichi Fukui analyzed the frontier orbitals of such systems.  A process in which the HOMO-LUMO interaction is constructive (results in a net bonding interaction) is favorable and considered symmetry-allowed, while a process in which the HOMO-LUMO interaction is non-constructive (results in bonding and antibonding interactions that cancel) is disfavorable and considered symmetry-forbidden. 
Though conceptually distinct, aromatic transition state theory (Zimmerman and Dewar), frontier molecular orbital theory (Fukui), and orbital symmetry conservation (Woodward and Hoffmann) all make identical predictions.  The Woodward–Hoffmann rules exemplify molecular orbital theory's power, and indeed helped demonstate that useful chemical results could arise from orbital analysis.  The discovery would earn Hoffmann and Fukui the 1981 Nobel Prize in Chemistry. By that time, Woodward had died, and so was ineligible for the prize.