The recent synthesis of single-layer Janus-type transition metal dichalcogenides (TMDs) raises the question of the existence of other possible 2D structures with an asymmetric out-of-plane structural configuration. In the present work, a theoretical search for new Janus structures having SMoSe and SVSe compositions is performed. A detailed crystal-chemical analysis of the predicted structures is carried out, and it is shown that some of the dynamically stable structures are characterized by crystal-chemical features that are unique among TMDs, including quadruple Mo—Mo bonds and covalent S—S and Se—Se bonds. It is also shown that Mo-bearing TMDs have a tendency to form strong Mo—Mo bonds with chains or isolated dimers of molybdenum atoms, while in the case of vanadium-containing TMDs this feature is not characteristic. Two predicted crystal structures, called 1M-SVSe and 1A′-SMoSe, are especially promising for experimental synthesis and practical applications owing to their dynamical stability and rather low value of enthalpy compared with known structures. The enthalpy of 1M-SVSe is 0.22 eV per formula unit lower than that of 1T-SVSe, while the enthalpy of 1A′-SMoSe is 0.12 eV per formula unit lower than the enthalpy of 1T-SMoSe. The performed topological analysis showed that the predicted structures are unique and do not have analogues in the Inorganic Crystal Structure Database.