  We prove that any group acting faithfully on a bifoliated plane while
preserving the orientations of both foliations is left-orderable. The proof
utilizes a construction of a linear order on the set of ends of the leaf
spaces, which takes advantage of the additional structure coming with a
bifoliation. Moreover, we build an identification between ends of leaf spaces
of the bifoliation and subsets of the boundary circle at infinity, and use it
to give a condition for the equivalence between a faithful group action on a
bifoliated plane and a faithful group action on the set of ends of the leaf
spaces.
