  Given a solution to 4D Einstein gravity with an isometry direction, it is
known that the equations of motion are identical to those of a 3D
$\sigma$-model with target space geometry $SU(1,1)/U(1)$. Thus, any
transformation by $SU(1, 1) \cong SL(2,\mathbb{R})$ is a symmetry for the
action and allows one to generate new solutions in 4D. Here we clarify and
extend recent work on electromagnetic (EM) duality in the context of the
classical double copy. In particular, for pure gravity, we identify an explicit
map between the Maxwell field of the single copy and the scalars in the target
space, allowing us to identify the $U(1) \subset SL(2, \mathbb{R})$ symmetry
dual to EM duality in the single copy. Moreover, we extend the analysis to
Einstein-Maxwell theory, where we highlight the role of Ehlers-Harrison
transformations and, for spherically symmetric charged black hole solutions, we
interpret the equations of motion as a truncation of the putative single copy
for Einstein-Yang-Mills theory.
