A linear stability analysis of laterally confined swirling flow is given, of the type described by Long's equation in the inviscid limit or by the von Kármán similarity equations in the absence of lateral confinement. The flow of interest involves identical counterflowing fluid streams injected with equal velocity W0 through opposing porous disks, rotating with angular velocities Ω and ±Ω, respectively, about a common normal axis. By means of mass transfer experiments on an aqueous system of this type we have detected an apparent hydrodynamic instability having the appearance of an inviscid supercritical bifurcation at a certain |Ω| > 0. As an attempt to elucidate this phenomenon, linear stability analyses are performed on several idealized flows, by means of a numerical Galerkin technique. An analysis of high-Reynolds-number similarity flow predicts oscillatory instability for all non-zero Ω. The spatial structure of the most unstable modes suggests that finite container geometry, as represented by the confining cylindrical sidewalls, may have a strong influence on flow stability. This is borne out by an inviscid stability analysis of a confined flow described by Long's equation. This analysis suggests a novel bifurcation of the inviscid variety, which serves qualitatively to explain the results of our mass transfer experiments.