  Interval exchange transformations are typically uniquely ergodic maps and
therefore have uniformly distributed orbits. Their degree of uniformity can be
measured in terms of the star-discrepancy. Few examples of interval exchange
transformations with low-discrepancy orbits are known so far and only for
$n=2,3$ intervals, there are criteria to completely characterize those interval
exchange transformations. In this paper, it is shown that having
low-discrepancy orbits is a conjugacy class invariant under composition of
maps. To a certain extent, this approach allows us to distinguish interval
exchange transformations with low-discrepancy orbits from those without. For
$n=4$ intervals, the classification is almost complete with the only
exceptional case having monodromy invariant $\rho = (4,3,2,1)$. This particular
monodromy invariant is discussed in detail.
