  We revisit the previously known local inversion symmetry of the
five-dimensional Kerr-AdS metric that relates the over-rotating black hole to
the under-rotating one and reinterpret it as an interchanging symmetry between
time and the longitudinal angular coordinates. We generalize this to all $D$
dimensions, including $D=4$, thereby enlarging the trivial linear $\mathbb Z_N$
symmetry of the $N=\lfloor(D-1)/2\rfloor$ longitudinal angular coordinates to
the nonlinearly realized $\mathbb Z_{N+1}$ symmetry that involves time.
