  We establish an equivalence between conformally Einstein--Maxwell Kahler
4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds
(in the sense of Calabi) with nowhere vanishing scalar curvature. The
corresponding pairs of Kahler metrics arise as transversal Kahler structures of
Sasaki metrics compatible with the same CR structure and having commuting
Sasaki--Reeb vector fields. This correspondence extends to higher dimensions
using the notion of a weighted extremal Kahler metric, illuminating and uniting
several explicit constructions in Kahler and Sasaki geometry. It also leads to
new existence and non-existence results for extremal Sasaki metrics, suggesting
a link between notions of relative weighted K-stability for a polarized
variety, and relative K-stability of the Kahler cone corresponding to a Sasaki
polarization.
