  We relate three classes of nonpositively curved metric spaces: hierarchically
hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We
show that every hierarchically hyperbolic space admits a new metric that is
coarsely injective. The new metric is quasi-isometric to the original metric
and is preserved under automorphisms of the hierarchically hyperbolic space. We
show that every coarsely injective metric space of uniformly bounded geometry
is strongly shortcut. Consequently, hierarchically hyperbolic groups --
including mapping class groups of surfaces -- are coarsely injective and
coarsely injective groups are strongly shortcut.
  Using these results, we deduce several important properties of hierarchically
hyperbolic groups, including that they are semihyperbolic, have solvable
conjugacy problem, have finitely many conjugacy classes of finite subgroups,
and that their finitely generated abelian subgroups are undistorted. Along the
way we show that hierarchically quasiconvex subgroups of hierarchically
hyperbolic groups have bounded packing.
