Zappa-Szép products arise when an algebraic structure has the property that every element has a unique decomposition as a product of elements from two given substructures. They may also be constructed from actions of two structures on one another, satisfying axioms first formulated by G. Zappa, and have a natural interpretation within automata theory. We study Zappa-Szép products arising from actions of a group and a band, and study the structure of the semigroup that results. When the band is a semilattice, the Zappa-Szép product is orthodox and -unipotent. We relate the construction (via automata theory) to the ?-semidirect product of inverse semigroups devised by Billhardt. © 2008 Springer Science+Business Media, LLC.