  The compositional as well as structural asymmetries in Janus transition metal
dichalcogenides (J-TMDs) and their van der Waals heterostructures (vdW HSs)
induce an intrinsic Rashba spin-splitting. We investigate the variation of
band-gaps and the Rashba parameter in three different Janus heterostructures
having AB-stacked Mo$XY$/W$XY$ ($X$, $Y$ = S, Se, Te; $X\neq Y$) geometry with
a $Y-Y$ interface, using first-principles calculations. We consider the effect
of external electric field and in-plane biaxial strain in tuning the strength
of the intrinsic electric field, which leads to remarkable modifications of the
band-gap and the Rashba spin-splitting. In particular, it is found that the
positive applied field and compressive in-plane biaxial strain can lead to a
notable increase in the Rashba spin-splitting of the valence bands about the
$\Gamma$-point. Moreover, our \textit{ab-initio} density functional theory
(DFT) calculations reveal the existence of a type-II band alignment in these
heterostructures, which remains robust under positive external field and
biaxial strain. These suggest novel ways of engineering the electronic,
optical, and spin properties of J-TMD van der Waals heterostructures holding a
huge promise in spintronic and optoelectronic devices. Detailed $\mathbf{k\cdot
p}$ model analyses have been performed to investigate the electronic and spin
properties near the $\Gamma$ and K points of the Brillouin zone.
