The particle orbits in field-reversed mirrors (FRM) are studied. It is found that the orbit can be expressed in terms of the first and third kind of elliptic integral for particles moving in the centre plane of the Hill vortex configuration. Requirements for absolute confinement are obtained, and four distinct classes of orbits are identified that are bounded by limits on the canonical angular momentum Pθ. The improvement in confinement over a simple mirror as well as natural divertor properties of the FRM are illustrated by these results.