AbstractInterval 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 $$n=2,3$$
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 $$n=4$$
n
=
4
intervals, the classification is almost complete with the only exceptional case having monodromy invariant $$\rho = (4,3,2,1)$$
ρ
=
(
4
,
3
,
2
,
1
)
. This particular monodromy invariant is discussed in detail.