  Reflection in a line in Euclidean 3-space defines an almost paracomplex
structure on the space of all oriented lines, isometric with respect to the
canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of
oriented geodesics of any real 3-dimensional space form admits both isometric
and anti-isometric paracomplex structures.
  This paper considers the existence or otherwise of isometric and
anti-isometric almost paracomplex structures $j$ on a pseudo-Riemannian
4-manifold $(M,g)$, such that $j$ is parallel with respect to the Levi-Civita
connection of $g$. It is shown that if an isometric or anti-isometric almost
paracomplex structure on a conformally flat manifold is parallel, then the
scalar curvature of the metric must be zero. In addition, it is found that $j$
is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal
foliations by totally geodesic surfaces.
  The composition of a Riemannian metric with an isometric almost paracomplex
structure $j$ yields a neutral metric $g'$. It is proven that if $j$ is
parallel, then $g$ is Einstein iff $g'$ is conformally flat and scalar flat.
  The vanishing of the Hirzebruch signature is found to be a necessary
topological condition for a closed 4-manifold to admit an Einstein metric with
a parallel isometric paracomplex structure. Thus, while the K3 manifold admits
an Einstein metric with an isometric paracomplex structure, it cannot be
parallel. The same holds true for certain connected sums of complex projective
2-space and its conjugate.
