In this paper we describe a novel magnetizing device based on eight rotatable permanent magnets arranged in a quadrupolar configuration, which is termed the TetraMag. TetraMag creates stable and homogeneous magnetic fields at the sample position with a resolution of 0.02 mT tunable between -570 mT and +570 mT. The field direction is continuously rotatable between 0° and 360° within the sample plane, while the field strength is maintained. A simplified mathematical description of TetraMag is developed leading to magnetic field calculations which are in good agreement with the experimental results. This versatile device avoids electrical energy dissipation, cooling mechanisms, and hysteresis effects known from classical electromagnets. It is ultrahigh vacuum compatible and it offers a completely free optical path over 180° for magneto-optical experiments. It is suitable for scattering experiments with synchrotron radiation and neutrons and may be employed in a large class of magnetization experiments.