RFeO3 orthoferrites, where R is a rare-earth ion of the lanthanide series, are attracting attention mostly because of their promising fast spin dynamics. The magnetic properties of these materials seem to crucially depend on whether the magnetizations of the R and Fe ions' weak ferromagnetic (WFM) components are parallel or antiparallel to each other. Here, we report an extensive investigation of a high-quality DyFeO3 single crystal in which the induced Dy3+ magnetization (FDy) has a natural tendency to be antiparallel to Fe3+ sublattice magnetization (FFe) within a large temperature window. Moreover, we find that specific variations of temperature and applied magnetic fields allow us to make FDy parallel to FFe, or force a spin-flip transition in FFe, among other effects. We found three different magnetic states that respond to temperature and magnetic fields, i.e. linear versus constant or, alternatively, presenting either behavior depending on the history of the sample. An original magnetic field-versus-temperature phase diagram is constructed to indicate the region of stability of the different magnetic phases, and to reveal the precise conditions yielding sudden spin switching and reversals. Knowledge of such a phase diagram is of potential importance to applications in spintronics and magnetic devices.