Let $\Sigma_{g,p}$ be the genus--$g$ oriented surface with $p$ punctures, with either $g>0$ or $p>3$. We show that $MCG(\Sigma_{g,p})/DT$ is acylindrically hyperbolic where $DT$ is the normal subgroup of the mapping class group $MCG(\Sigma_{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma_{g,p}$ for suitable $K$. Moreover, we show that in low complexity $MCG(\Sigma_{g,p})/DT$ is in fact hyperbolic. In particular, for $3g-3+p\leq 2$, we show that the mapping class group $MCG(\Sigma_{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma_{g,p})$ is separable. The aforementioned results follow from general theorems about composite rotating families that come from a collection of subgroups of vertex stabilisers for the action of a group $G$ on a hyperbolic graph $X$. We give conditions ensuring that the graph $X/N$ is again hyperbolic and various properties of the action of $G$ on $X$ persist for the action of $G/N$ on $X/N$.