The arrangement of spins at interfaces in a layered magnetic material often has an important effect on the properties of the material. One example of this is the directional coupling between the spins in an antiferromagnet and those in an adjacent ferromagnet, an effect first discovered in 1956 and referred to as exchange bias. Because of its technological importance for the development of advanced devices such as magnetic read heads and magnetic memory cells, this phenomenon has received much attention. Despite extensive studies, however, exchange bias is still poorly understood, largely due to the lack of techniques capable of providing detailed information about the arrangement of magnetic moments near interfaces. Here we present polarization-dependent X-ray magnetic dichroism spectro-microscopy that reveals the micromagnetic structure on both sides of a ferromagnetic-antiferromagnetic interface. Images of thin ferromagnetic Co films grown on antiferromagnetic LaFeO3 show a direct link between the arrangement of spins in each material. Remanent hysteresis loops, recorded for individual ferromagnetic domains, show a local exchange bias. Our results imply that the alignment of the ferromagnetic spins is determined, domain by domain, by the spin directions in the underlying antiferromagnetic layer.