We study the dynamics of a viscous fluid confined within an elastic pipe that is being stretched and twisted in the axial and azimuthal directions, respectively. A comprehensive numerical exploration of self-similar flows has been conducted for a wide range of stretching and twisting acceleration rates. Up to eight different families of steady flows have been identified and tracked. For purely axial stretching, the presented computations recover the results obtained by former authors in the past. However, the twisting mechanism, seen as a homotopy parameter, has been crucial in this study to explain how different families of equilibria interact and azimuthal swirl is induced. The explorations conclude that stable time-periodic flows may also take over the dynamics when the pipe is contracting in the axial direction, with mild or absent azimuthal twist. These time-periodic flows may destabilize through different codimension-two bifurcation scenarios, sometimes leading to unstable tori that have been genuinely identified by means of edge-tracking techniques.