  Linear sets on the projective line have attracted a lot of attention because
of their link with blocking sets, KM-arcs and rank-metric codes. In this paper,
we study linear sets having two points of complementary weight, that is with
two points for which the sum of their weights equals the rank of the linear
set. As a special case, we study those linear sets having exactly two points of
weight greater than one, by showing new examples and studying their equivalence
issue. Also we determine some linearized polynomials defining the linear sets
recently introduced by Jena and Van de Voorde (2021).
