The reaction of an aryne with an alkyne to generate a benzocyclobutadiene (BCB) intermediate is rare. We report here examples of this reaction, revealed by Diels-Alder trapping of the BCB by either pendant or external electron-deficient alkynes. Mechanistic delineation of the reaction course is supported by DFT calculations. A three-component process joining the benzyne first with an electron-rich and then with an electron-poor alkyne was uncovered. Reactions in which the BCB functions in a rarely observed role as a 4π diene component in Diels-Alder reactions are reported. The results also shed new light on aspects of the hexadehydro-Diels-Alder reaction used to generate the benzynes.