Arrays of nanomagnets have important potential applications as future generation ultrahigh-density patterned magnetic recording media, in which each nanomagnet constitutes a single bit. We introduce a powerful technique to identify and quantify reversible and irreversible magnetization changes, a key challenge in characterizing these systems. The experimental protocol consists of measuring a few families of second-order reversal curves along selected profiles in the first-order-reversal-curve diagram, which then can be decomposed into truly irreversible switching events and reversible magnetization changes. The viability of the method is demonstrated for arrays of sub-100-nm Fe nanomagnets, which exhibit complex magnetization reversal processes.