A spray dispenser is configured to generate a swirled output spray pattern 152 with improved rotating or angular velocity ω and smaller sprayed droplet size. Cup-shaped nozzle member 60 has a cylindrical side wall 62 surrounding a central longitudinal axis 64 and has a circular closed end wall 68 with at least one exit aperture 74 passing through the end wall. At least one enhanced swirl inducing mist generating structure is formed in an inner surface 70 of the end wall, and including a pair of opposed inwardly tapered offset power nozzle channels 80, 82 terminating in an interaction chamber 84 surrounding the exit aperture 74. The power nozzle channels generate opposing offset flows which are aimed to very efficiently generate a vortex of fluid which projects distally from the exit aperture as a swirled spray of small droplets 152 having a rapid angular velocity.