A vector-dominance two-photon exchange model is proposed to explain the recently observed production of $\rho^0\rho^0$ and $\rho^0\phi$ pairs in $e^+e^-$ annihilation at 10.58 GeV with the BaBar detector. All the observed features of the data --angular and decay distributions, rates-- are in agreement with the model. Predictions are made for yet-unobserved final states.