  Given a finite poset P, we consider pairs of linear extensions of P with
maximal distance, where the distance between two linear extensions L_1, L_2 is
the number of pairs of elements of P appearing in different orders in L_1 and
L_2. A diametral pair maximizes the distance among all pairs of linear
extensions of P. Felsner and Reuter defined the linear extension diameter of P
as the distance between a diametral pair of linear extensions.
  We show that computing the linear extension diameter is NP-complete in
general, but can be solved in polynomial time for posets of width 3.
  Felsner and Reuter conjectured that, in every diametral pair, at least one of
the linear extensions reverses a critical pair. We construct a counterexample
to this conjecture. On the other hand, we show that a slightly stronger
property holds for many classes of posets: We call a poset "diametrally
reversing" if, in every diametral pair, both linear extensions reverse a
critical pair. Among other results we show that interval orders and 3-layer
posets are diametrally reversing. From the latter it follows that almost all
posets are diametrally reversing.
