We provide combinatorial interpretations of a pair of dual identities, one satisfied by each kind of Stirling number. The first identity is generated by partitioning all of the k-cycle permutations of [n] acording to which elements of [n] are cycle minima; the second identity is obtained by dividing the k-calss partitions of [n] according to which elements are class minima. These identities are two instances of a more general identity, which applies to all families of binomial type. We establish the more general identity and give further instances of it.