The development of new synthetic methodologies is instrumental for enabling the discovery of new medicines. The methods that provide efficient access to structural alternatives for aromatic compounds (that is, saturated arene bioisosteres) have become highly coveted1–4. The incorporation of these bioisosteres typically leads to favourable drug-like properties and represents an emerging field of research. Here we report a new synthetic method that furnishes a coveted motif, the bicyclo[2.1.1]hexane scaffold5,6, using mild reaction conditions and an operationally simple protocol. The methodology proceeds through the uncommon coupling of two strained fragments: transiently generated cyclic allenes and bicyclo[1.1.0]butanes, which possess considerable strain energies of about 30 kcal mol−1 (ref. 7) and about 60 kcal mol−1 (ref. 6), respectively. The reaction is thought to proceed by a σ-bond insertion through a diradical pathway. However, rather than requiring an external stimulus to generate radical species, reactivity is thought to arise as a result of innate diradical character present in each reactant. This diradicaloid character8, an underused parameter in reaction design, arises from the severe geometric distortions of each reactant. Our studies provide a means to access functionalized bicyclo[2.1.1]hexanes of value for drug discovery, underscore how geometric distortion of reactants can be used to enable uncommon modes of reactivity and should encourage the further exploration and strategic use of diradicaloids in chemical synthesis.