  The new class of Janus two-dimensional (2D) transition-metal dichalcogenides
with two different interfaces are currently gaining increasing attention due to
their distinct properties different from the typical 2D materials. Here, we
show that in-plane anisotropy of a 2D atomic crystal, like ReS$_{2}$ or
ReSe$_{2}$, allows formation of a large number of inequivalent Janus
monolayers. We use first-principles calculations to investigate the structural
stability of 29 distinct ReX$_{2-x}$Y$_{x}$ ($\mathrm{X,Y \in \{S,Se\}}$)
structures, which can be obtained by selective exchange of exposed chalcogens
in a ReX$_{2}$ monolayer. We also examine the electronic properties and work
function of the most stable Janus monolayers and show that the large number of
inequivalent structures provides a way to engineer spin-orbit splitting of the
electronic bands. We find that the breaking of inversion symmetry leads to
sizable spin splittings and spontaneous diople moments than are larger than
those in other Janus dichalcogenides. Moreover, our caluclations suggest that
the work function of the Janus monolayers can be tuned by varying the content
of the substituting chalcogen. Our work demonstrates that in-plane anisotropy
provides additional flexibility in sub-layer engineering of 2D atomic crystals.
