The generalized Woodward–Hoffmann rules, first given in 1969, are equivalent to an earlier general approach, the Möbius-Hückel concept of Zimmerman, which was first stated in 1966 and is also known as aromatic transition state theory. As its central tenet, aromatic transition state theory holds that allowed pericyclic reactions proceed via transition states with aromatic character, while forbidden pericyclic reactions would encounter transition states that are antiaromatic in nature.  In the Dewar-Zimmerman analysis, one is concerned with the topology of the transition state of the pericyclic reaction.  If the transition state involves 4n electrons, the Möbius topology is aromatic and the Hückel topology is antiaromatic, while if the transition state involves 4n + 2 electrons, the Hückel topology is aromatic and the Möbius topology is antiaromatic.  The parity of the number of phase inversions (described in detail below) in the transition state determines its topology.  A Möbius topology involves an odd number of phase inversions whereas a Hückel topology involves an even number of phase inversions.
In connection with Woodward–Hoffmann terminology, the number of antarafacial components and the number of phase inversions always have the same parity. Consequently, an odd number of antarafacial components gives Möbius topology, while an even number gives Hückel topology. Thus, to restate the results of aromatic transition state theory in the language of Woodward and Hoffmann, a 4n-electron reaction is thermally allowed if and only if it has an odd number of antarafacial components (i.e., Möbius topology); a (4n + 2)-electron reaction is thermally allowed if and only if it has an even number of antarafacial components (i.e., Hückel topology).
Procedure for Dewar-Zimmerman analysis (examples shown on the right): Step 1.  Shade in all basis orbitals that are part of the pericyclic system.  The shading can be arbitrary.  In particular the shading does not need to reflect the phasing of the polyene MOs; each basis orbital simply need to have two oppositely phased lobes in the case of p or sp hybrid orbitals, or a single phase in the case of an s orbital.  Step 2. Draw connections between the lobes of basis orbitals that are geometrically well-disposed to interact at the transition state.  The connections to be made depend on the transition state topology.  (For example, in the figure, different connections are shown in the cases of con- and disrotatory electrocyclization.)  Step 3.  Count the number of connections that occur between lobes of opposite shading: each of these connections constitutes a phase inversion.  If the number of phase inversions is even, the transition state is Hückel, while if the number of phase inversions is odd, the transition state is Möbius.  Step 4.  Conclude that the pericyclic reaction is allowed if the electron count is 4n + 2 and the transition state is Hückel, or if the electron count is 4n and the transition state is Möbius; otherwise, conclude that the pericyclic reaction is forbidden.
Importantly, any scheme of assigning relative phases to the basis orbitals is acceptable, as inverting the phase of any single orbital adds 0 or ±2 phase inversions to the total, an even number, so that the parity of the number of inversions (number of inversions modulo 2) is unchanged.