In a conrotatory mode of an electrocyclic reaction (a class of organic chemical reactions) the substituents located at the termini of a conjugated double bond system move in the same (clockwise or counter clockwise) direction during ring opening or ring closure. In a disrotatory mode, they move in opposite directions.
A simple arrow-pushing analysis fails to explain the quantitative conversion of trans-cis-trans-2,4,6-octatriene to cis-dimethylcyclohexadiene (top of figure). However, a detailed look at the orbital mechanics behind the process reveals that the orbital symmetry of the octatriene's HOMO requires for the end pi orbitals to move in opposite directions to form the correct symmetry found in the sigma bond.
This stereospecificity is found in thermal rearrangements of all conjugated systems containing 4n + 2 pi electron and is based on preservation of orbital symmetry in the highest occupied molecular orbital. Systems containing 4n pi electrons show the opposite conrotatory mode, as do photoinduced rearrangements of 4n + 2 pi electrons. Photoinduced rearrangements of 4n electron systems follow the disrotatory rule.
The Woodward-Hoffmann rules summarize the above different reactions.