  In this paper we study the left-orderability of $3$-manifold groups using an
enhancement, called recalibration, of Calegari and Dunfield's "flipping"
construction, used for modifying $\mbox{Homeo}_+(S^1)$-representations of the
fundamental groups of closed $3$-manifolds. The added flexibility accorded by
recalibration allows us to produce $\mbox{Homeo}_+(S^1)$-representations of
hyperbolic link exteriors so that a chosen element in the peripheral subgroup
is sent to any given rational rotation. We apply these representations to show
that the branched covers of families of links associated to arbitrary
epimorphisms of the link group onto a finite cyclic group are left-orderable.
This applies, for instance, to fibered hyperbolic strongly quasipositive links.
Our result on the orderability of branched covers implies that the degeneracy
locus of any pseudo-Anosov flow on an alternating knot complement must be
meridional, which generalizes the known result that the fractional Dehn twist
coefficient of any hyperbolic fibered alternating knot is zero. Applications of
these representations to order-detection of slopes are also discussed in the
paper.
