We studied the membership, size ratio and rotational properties of 13 asteroid clusters consisting of between 3 and 19 known members that are on similar heliocentric orbits. By backward integrations of their orbits, we confirmed their cluster membership and estimated times elapsed since separation of the secondaries (the smaller cluster members) from the primary (i.e., cluster age) that are between 105 and a few 106 years. We ran photometric observations for all the cluster primaries and a sample of secondaries and we derived their accurate absolute magnitudes and rotation periods. We found that 11 of the 13 clusters follow the same trend of primary rotation period vs mass ratio as asteroid pairs that was revealed by Pravec et al. (2010). We generalized the model of the post-fission system for asteroid pairs by Pravec et al. (2010) to a system of N components formed by rotational fission and we found excellent agreement between the data for the 11 asteroid clusters and the prediction from the theory of their formation by rotational fission. The two exceptions are the high-mass ratio (q > 0.7) clusters of (18777) Hobson and (22280) Mandragora for which a different formation mechanism is needed. Two candidate mechanisms for formation of more than one secondary by rotational fission were published: the secondary fission process proposed by Jacobson and Scheeres (2011) and a cratering collision event onto a nearly critically rotating primary proposed by Vokrouhlický et al. (2017). It will have to be revealed from future studies which of the clusters were formed by one or the other process. To that point, we found certain further interesting properties and features of the asteroid clusters that place constraints on the theories of their formation, among them the most intriguing being the possibility of a cascade disruption for some of the clusters.