Kuperberg's SL3 web basis is importantly rotation invariant, meaning that rotations of basis webs are themselves basis webs. In recent work, we extended this construction to a rotation invariant basis of SL4 webs. The guiding principle is that basis webs should naturally biject with tableaux in a way that intertwines web rotation with tableau promotion, while trips in webs (in the sense of Postnikov's plabic graphs) should correspond with promotion paths in tableaux. The main problem is to extend this framework to arbitrary rank. We show partial progress in this direction. In particular, we extend the correspondence to all two-column tableaux and separately to all acyclic webs. Further confirmation of the naturality of these constructions comes from connections to the geometry of corresponding Springer fibres. This talk is based on work with Ron Cherny, Mike Cummings, Christian Gaetz, Stephan Pfannerer, Jessica Striker, and Josh Swanson.