  We examine a generalisation of the usual self-duality equations for
Yang-Mills theory when the colour space admits a non-trivial involution. This
involution allows us to construct a non-trivial twist which may be combined
with the Hodge star to form a twisted self-dual curvature. We will construct a
simple example of twisted self-duality for $su(2) \oplus su(2)$ gauge theory
along with its explicit solutions and then dimensionally reduce from four
dimensions to obtain families of non-trivial non-linear equations in lower
dimensions. This twisted self-duality constraint will be shown to arise in E_7
exceptional field theory through a Scherk-Schwarz reduction and we will show
how an Eguchi-Hanson gravitational instanton also obeys the twisted
self-duality condition.
