A population of small main-belt asteroids residing in pairs with closely similar orbits was revealed by Vokrouhlicky and Nesvorny (2008). As a part of an international collaboration, we have performed a photometric observing campaign to characterize the spin properties of these pairs from which we examine a possible common origin of each pair. Observations of ~30 primaries (larger members of asteroid pairs) reveal a correlation between the rotation periods of primaries and the secondary-to-primary size ratio: primaries of pairs with small secondaries rotate rapidly, while primaries of pairs with large secondaries show significantly slower rotations. This observed correlation is consistent with a proposed formation mechanism of disrupted asteroids by a rotational fission of the parent body or a gentle split of a binary pair. The spin rate of the primary is reduced while a part of its rotational angular momentum is transferred to the relative motion of the system's components during separation. This is yet another observational evidence for the porous and shattered nature of asteroids' structure.