  Freudenthal duality (FD) is a non-linear symmetry of the Bekenstein-Hawking
entropy of extremal dyonic black holes (BHs) in Maxwell-Einstein-scalar
theories in four space-time dimensions realized as an anti-involutive map in
the symplectic space of electric-magnetic BH charges. In this paper, we
generalize FD to the class of rotating (stationary) extremal BHs, both in the
under- and over-rotating regime, defining a (generalized) rotating FD
(generally, non-anti-involutive) map (RFD), which also acts on the BH angular
momentum. We prove that the RFD map is unique, and we compute the explicit
expression of its non-linear action on the angular momentum itself.
Interestingly, in the non-rotating limit, RFD bifurcates into the usual,
non-rotating FD branch and into a spurious branch, named "golden" branch,
mapping a non-rotating (static) extremal BH to an under-rotating (stationary)
extremal BH, in which the ratio between the angular momentum and the
non-rotating entropy is the square root of the golden ratio. Finally, we
investigate the possibility of inducing transitions between the under- and
over-rotating regimes by means of RFD, obtaining a no-go result.
