We study the phase diagram and other general macroscopic properties of an interacting spin (or particle) system out of equilibrium, namely, a reaction-diffusion Ising model whose time evolution occurs as a consequence of a combination of spin-flip (Glauber) and spin-exchange (Kawasaki) processes. The Glauber rate at site x when the configuration is s, say c(s;x), satisfies detailed balance at a reciprocal temperature β, while the Kawasaki rate for the interchange between nearest-neighbor sites x and y, Γc(s;x,y), satisfies detailed balance at temperature β’. We derive hydrodynamic-type macroscopic equations from the stochastic microscopic model for β’,β≥0 and large Γ when time and space are rescaled by Γ and √Γ , respectively, and study the homogeneous steady solutions of those equations when Γ→∞. We state some general theorems for β’=0 and solve explicitly the model with different choices c(s;x) for systems of arbitrary dimension d when β’=0 and also for d=1 when β’≠0. We also describe new Monte Carlo data for finite Γ, β’=0, and d=1,2. The latter suggests, in particular, the existence of phase transitions for d=1, finite Γ, and some choices for c(s;x).