  We study quotients of mapping class groups of punctured spheres by suitable
large powers of Dehn twists, showing an analogue of Ivanov's theorem for the
automorphisms of the corresponding quotients of curve graphs. Then we use this
result to prove quasi-isometric rigidity of these quotients, answering a
question of Behrstock, Hagen, Martin, and Sisto in the case of punctured
spheres. Finally, we show that the automorphism groups of our quotients of
mapping class groups are "small", as are their abstract commensurators. This is
again an analogue of a theorem of Ivanov about the automorphism group of the
mapping class group. In the process we develop techniques to extract
combinatorial data from a quasi-isometry of a hierarchically hyperbolic space,
and use them to give a different proof of a result of Bowditch about
quasi-isometric rigidity of pants graphs of punctured spheres.
