A computational study is presented using a sequence of full-potential linearized augmented plane-wave method (FP-LAPW) within the generalized gradient approximation as well as the on-site hybrid functionals for the exchange–correlation energy to determine the structural and electronic properties of Janus transition metal dichalcogenide monolayers in MoXY (X, Y = S, Se, Te with X different from Y) configurations. The calculated electronic band structures of the studied Janus monolayers show a Rashba splitting around the Γ point and the Zeeman spin-splitting at the K₋ and K₊ points. The findings suggest that these materials represent interesting 2D systems to develop different applications such as orbitronics.