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The behavior and rheology of a dispersion of Janus drops (or Janus emulsion) under a steady shear flow are explored in the infinite dilution limit. To achieve analytical progress, the Janus drops are assumed to consist of a pair of fluids bounded to hemispherical domains of equal radii. At ‘freely’ suspended conditions the Janus drops undergo periodic orbits in a shear flow that are intermediate to that of a solid sphere and a disk that depend on the viscosities of the internal fluids. Non-Newtonian behavior is found for this system on account of the anisotropic hydrodynamics of the Janus drops. The viscosity of the Janus emulsion that corresponds to the minimum energy of dissipation is analogous to that derived by Taylor (1932) for a dispersion of simple drops. It is also found that an external force can induce the Janus drops to adopt a preferential orientation in a shear flow. Interestingly, a neutrally buoyant Janus drop with a displaced center of gravity can migrate lateral to the undisturbed shear flow; it is inferred that this phenomenon can lead to spatial-dependent rheology in pressure-driven flows. [ABSTRACT FROM AUTHOR]