Over the past several years, a significant effort has been focused toward the fabrication of nano-scale ``Janus'' particles with patterned surface properties. In this paper, building on the recent work of Swan and Khair, we investigate a simple model system for a hydrodynamically Janus particle: the "slip-stick" sphere, whose surface is partitioned into two distinct regions. On one region, fluid adjacent to the particle obeys the no-slip condition, whereas on the other, fluid slips past the particle. The Janus nature of a slip-stick sphere leads to a number of interesting and somewhat surprising results when it is placed in different flows. For example, in a pure straining field the sphere attains an equilibrium orientation either along the compressional or extensional axis of the flow, depending on the ratio of slip-to-stick surface areas. In a simple shear flow, the slip-stick sphere undergoes a periodic rotational motion, or Jeffrey orbit. Moreover, depending on its initial orientation, the particle can either follow a periodic translational orbit or undergo a net displacement along the flow direction. Lastly, to first order in the volume fraction of slip-stick spheres, the suspension rheology is non-Newtonian, with non-zero first and second normal stress differences.