Longuet-Higgins and E. W. Abrahamson showed that the Woodward–Hoffmann rules can best be derived by examining the correlation diagram of a given reaction.  A symmetry element is a point of reference (usually a plane or a line) about which an object is symmetric with respect to a symmetry operation.  If a symmetry element is present throughout the reaction mechanism (reactant, transition state, and product), it is called a conserved symmetry element.  Then, throughout the reaction, the symmetry of molecular orbitals with respect to this element must be conserved.  That is, molecular orbitals that are symmetric with respect to the symmetry element in the starting material must be correlated to (transform into) orbitals symmetric with respect to that element in the product.  Conversely, the same statement holds for antisymmetry with respect to a conserved symmetry element.  A molecular orbital correlation diagram correlates molecular orbitals of the starting materials and the product based upon conservation of symmetry.  From a molecular orbital correlation diagram one can construct an electronic state correlation diagram that correlates electronic states (i.e. ground state, and excited states) of the reactants with electronic states of the products.  Correlation diagrams can then be used to predict the height of transition state barriers.
Although orbital "symmetry" is used as a tool for sketching orbital and state correlation diagrams, the absolute presence or absence of a symmetry element is not critical for the determination of whether a reaction is allowed or forbidden.  That is, the introduction of a simple substituent that formally disrupts a symmetry plane or axis (e.g., a methyl group) does not generally affect the assessment of whether a reaction is allowed or forbidden. Instead, the symmetry present in an unsubstituted analog is used to simplify the construction of orbital correlation diagrams and avoid the need to perform calculations.  Only the phase relationships between orbitals are important when judging whether a reaction is "symmetry"-allowed or forbidden. Moreover, orbital correlations can still be made, even if there are no conserved symmetry elements (e.g., 1,5-sigmatropic shifts and ene reactions). For this reason, the Woodward–Hoffmann, Fukui, and Dewar–Zimmerman analyses are equally broad in their applicability, though a certain approach may be easier or more intuitive to apply than another, depending on the reaction one wishes to analyze.