  We give an explicit local classification of conformally equivalent but
oppositely oriented Kaehler metrics on a 4-manifold which are toric with
respect to a common 2-torus action. In the generic case, these structures have
an intriguing local geometry depending on a quadratic polynomial and two
arbitrary functions of one variable, these two functions being explicit degree
4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi).
  One motivation for and application of this result is an explicit local
description of Einstein 4-manifolds which are hermitian with respect to either
orientation. This can be considered as a riemannian analogue of a result in
General Relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a
natural extension of the classification of selfdual Einstein hermitian
4-manifolds, obtained independently by R. Bryant and the first and third
authors.
  We discuss toric compactifications of these metrics on orbifolds and provide
infinite discrete families of compact toric extremal Kaehler orbifolds. Our
examples include Bach-flat Kaehler orbifolds which are conformal to complete
smooth Einstein metrics on an open subset. We illustrate how these examples fit
with recent conjectures relating the existence of extremal toric metrics to
various notions of stability.
