  We construct a twin correspondence between graphs with prescribed mean
curvature in three-dimensional Riemannian Killing submersions and spacelike
graphs with prescribed mean curvature in three-dimensional Lorentzian Killing
submersions. Our duality extends the Calabi correspondence between minimal
graphs in the Euclidean space $\mathbb{R}^3$ and maximal graphs in the
Lorentz-Minkowski spacetime $\mathbb{L}^3$, by allowing arbitrary prescribed
mean curvature and bundle curvature. For instance, we transform the prescribed
mean curvature equation in $\mathbb{L}^3$ into the minimal surface equation in
the generalized Heisenberg space with prescribed bundle curvature. We present
several applications of the twin correspondence to the study of the moduli
space of complete spacelike surfaces in certain Lorentzian spacetimes.
