In Part 1 of this series conservation principles for ring circulation and kinematic swirl angular momentum were developed for general axisymmetric incompressible viscous flow. These principles were then used to classify the four independent axial causes of swirl-free conically similar viscous flow. Part 2 provided a detailed analysis of the one-parameter swirl-free flows that are generated by each one of the axial singularities acting alone. The present paper extends, to swirling flow, the description of the axial singularities that drive axisymmetric viscous flow. In the special case of conically similar viscous flow, two independent half-line sources of swirl angular momentum suffice to complete the set of axial singularities that can generate such swirling flows. The individual strengths of the six independent axial causes provide a complete characterization of all conically similar viscous flows that can be generated in this way. This Part 3 completes the task of analysing in detail the independent one-parameter flows generated by axial causes by studying the flow caused by uniform production of kinematic swirl angular momentum on a half-axis. This flow demonstrates how swirl may induce an axial half-plane flow. For large swirl circulation strengths, swirl angular momentum diffuses and convects so as to fill slightly more than half the space with an almost constant density of swirl angular momentum. A well-developed internal boundary layer, in the form of an outward radial jet, then separates this region from one in which the flow is almost irrotational. The jet entrains two impinging convection fields. The angular location of the jet is determined by relating the axial component of moment of whirl produced at the origin to the strength of the swirling circulation singularity on the axis.