Recent advances in polymer chemistry allow a facile, large-scale synthesis of nanoscale Janus particles (JP) with tunable structural and physical properties. Both the structures and distributions of regions with different chemical compositions within JP play an important role in chemical and optical sensing, or in bio-medical applications, such as drug delivery. The structural properties of symmetric JP can be accurately characterized by small-angle scattering (SAS), yet the structure of JP with tunable geometrical and chemical asymmetries (AJP) can be described only qualitatively (e.g., globular, elongated or planar), depending on the value of the scattering exponent in the Porod region of SAS intensity. Here it is shown that identification of AJP and a quantitative description of their morphology can be achieved by using the method of SAS together with contrast variation. This approach is illustrated by providing analytic expressions for SAS intensities and for contrast matching points for two kinds of common multiphase AJP: spheres with one cap and those with two caps. The influence of the model's parameters is presented and discussed, and the structural evolution of AJP upon solvent deuteration is characterized. The results suggest that the combination of the SAS technique with multiphase modeling provides unprecedented detailed information about the structural conformation of AJP, which allows their identification from experimental SAS data. Monte Carlo simulations are performed both to validate the obtained results and to illustrate the above findings for complex AJP for which analytic expressions are not available.