  A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant,
states that compact symplectic toric orbifolds are classified by their moment
polytopes, together with a positive integer label attached to each of their
facets. In this paper we use this result, and the existence of "global"
action-angle coordinates, to give an effective parametrization of all
compatible toric complex structures on a compact symplectic toric orbifold, by
means of smooth functions on the corresponding moment polytope. This is
equivalent to parametrizing all toric Kahler metrics and generalizes an
analogous result for toric manifolds.
  A simple explicit description of interesting families of extremal Kahler
metrics, arising from recent work of R.Bryant, is given as an application of
the approach in this paper. The fact that in dimension four these metrics are
self-dual and conformally Einstein is also discussed. This gives rise in
particular to a one parameter family of self-dual Einstein metrics connecting
the well known Eguchi-Hanson and Taub-NUT metrics.
