  The search for a geometrical understanding of dualities in string theory, in
particular T-duality, has led to the development of modern T-duality covariant
frameworks such as Double Field Theory, whose mathematical structure can be
understood in terms of generalized geometry and, more recently, para-Hermitian
geometry. In this work we apply techniques associated to this doubled geometry
to four-dimensional manifolds, and we show that they are particularly
well-suited to the analysis of integrability in special spacetimes, especially
in connection with Penrose's twistor theory and its applications to general
relativity. This shows a close relationship between some of the geometrical
structures in the para-Hermitian approach to double field theory and those in
algebraically special solutions to the Einstein equations. Particular results
include the classification of four-dimensional, possibly complex-valued,
(para-)Hermitian structures in different signatures, the Lie and Courant
algebroid structures of special spacetimes, and the analysis of deformations of
(para-)complex structures. We also discuss a notion of "weighted algebroids" in
relation to a natural gauge freedom in the framework. Finally, we analyse the
connection with two- and three-dimensional (real and complex) twistor spaces,
and how the former can be understood in terms of the latter, in particular in
terms of twistor families.
