A series of nine 1,3-dipoles, belonging to the families of diazonium betaines, nitrilium betaines, and azomethine betaines, has been studied by means of the breathing-orbital valence bond ab initio method. Each 1,3-dipole is described as a linear combination of three valence bond structures, two zwitterions and one diradical, for which the weights in the total wave function can be quantitatively estimated. In agreement with an early proposition of Harcourt, the diradical character of 1,3-dipoles is shown to be a critical feature to favor 1,3-dipolar cycloaddition. Within each family, a linear relationship is evidenced between the weight of the diradical structure in the 1,3-dipole and the barrier to cycloaddition to ethylene or acetylene, with correlation coefficients of 0.98-1.00. The barrier heights also correlate very well with the transition energies from ground state to pure diradical states of the 1,3-dipoles at equilibrium geometry. Moreover, the weight of the diradical structure is shown to increase significantly in all 1,3-dipoles from their equilibrium geometries to their distorted geometries in the transition states. A mechanism for 1,3-dipolar cycloaddition is proposed, in which the 1,3-dipole first distorts so as to reach a reactive state that possesses some critical diradical character and then adds to the dipolarophile with little or no barrier. This mechanism is in line with the recently proposed distortion/interaction energy model of Ess and Houk and their finding that the barrier heights for the cycloaddition or a given 1,3-dipole to ethylene and acetylene are nearly the same, despite the exothermicity difference (Ess, D. H.; Houk, K. N. J. Am. Chem. Soc. 2008, 130, 10187).