This paper explains hysteretic transitions in swirling jets and models external flows of vortex suction devices. Toward this goal, the steady rotationally symmetric motion of a viscous incompressible fluid above an infinite conical stream surface of a half-angle θc is studied. The flows analysed are generalizations of Long's vortex. They correspond to the conically similar solutions of the Navier-Stokes equations and are characterized by circulation Γc given at the surface and axial flow force J1. Asymptotic analysis and numerical calculations show that four (for θc ≤ 90°) or five (for θc > 90°) solutions exist in some range of Γc and J1. The solution branches form hysteresis loops which are related to jump transitions between various flow regimes. Four kinds of jump are found : (i) vortex breakdown which transforms a near-axis jet into a two-cell flow with a reverse flow near the axis and an annular jet fanning out along conical surface θ = θs < θc ; (ii) vortex consolidation causing a reversal of (i) ; (iii) jump flow separation from surface θ = θc; and (iv) jump attachment of the swirling jet to the surface. As Γc and/or J1 decrease, the hysteresis loops disappear through a cusp catastrophe. The physical reasons for the solution non-uniqueness are revealed and the results are discussed in the context of vortex breakdown theories. Vortex breakdown is viewed as a fold catastrophe. Two new striking effects are found: (i) there is a pressure peak of O(Γc2) inside the annular swirling jet ; and (ii) a consolidated swirling jet forms with a reversed ('anti-rocket') flow force.