Amorphous solids, such as glasses, have complex responses to deformations, with substantial consequences in material design and applications. In this respect, two intertwined aspects are important: stability and reversibility. It is crucial to understand, on the one hand, how a glass may become unstable due to increased plasticity under shear deformations, and, on the other hand, to what extent the response is reversible, meaning how much a system is able to recover the original configuration once the perturbation is released. Here, we focus on assemblies of hard spheres as the simplest model of amorphous solids such as colloidal glasses and granular matter. We prepare glass states quenched from equilibrium supercooled liquid states, which are obtained by using the swap Monte Carlo algorithm and correspond to a wide range of structural relaxation time scales. We exhaustively map out their stability and reversibility under volume and shear strains using extensive numerical simulations. The region on the volume-shear strain phase diagram where the original glass state remains solid is bounded by the shear yielding and the shear jamming lines that meet at a yielding-jamming crossover point. This solid phase can be further divided into two subphases: the stable glass phase, where the system deforms purely elastically and is totally reversible, and the marginal glass phase, where it experiences stochastic plastic deformations at mesoscopic scales and is partially irreversible. The details of the stability-reversibilitymap depend strongly on the quality of annealing of the glass. This study provides a unified framework for understanding elasticity, plasticity, yielding, and jamming in amorphous solids.