  In the Teichm\"uller space of a hyperbolic surface of finite type, we
construct geodesic lines for Thurston's asymmetric metric having the property
that when they are traversed in the reverse direction, they are also geodesic
lines (up to reparametrization). The lines we construct are special stretch
lines in the sense of Thurston. They are directed by complete geodesic
laminations that are not chain-recurrent, and they have a nice description in
terms of Fenchel-Nielsen coordinates. At the basis of the construction are
certain maps with controlled Lipschitz constants between right-angled
hyperbolic hexagons having three non-consecutive edges of the same size. Using
these maps, we obtain Lipschitz-minimizing maps between hyperbolic particular
pairs of pants and, more generally, between some hyperbolic sufaces of finite
type with arbitrary genus and arbitrary number of boundary components. The
Lipschitz-minimizing maps that we contruct are distinct from Thurston's stretch
maps.
