The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is wellestablished, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected 'freezing by heating.' These phenomena are explored through systematic numerical simulations. This study reveals complicated relaxation processes as well as a crossover between two very different steady state regimes which are found to be bistable within a certain parameter range.