  We show that the rotating generalization of Hayward's non-singular black hole
previously studied in the literature is geodesically incomplete, and that its
straightforward extension leads to a singular spacetime. We present another
extension, which is devoid of any curvature singularity. The obtained metric
depends on three parameters and, depending on their values, yields an event
horizon or not. These two regimes, named respectively regular rotating Hayward
black hole and naked rotating wormhole, are studied both numerically and
analytically. In preparation for the upcoming results of the Event Horizon
Telescope, the images of an accretion torus around Sgr A*, the supermassive
object at the center of the Galaxy, are computed. These images contain, even in
the absence of a horizon, a central faint region which bears a resemblance to
the shadow of Kerr black holes and emphasizes the difficulty of claiming the
existence of an event horizon from the analysis of strong-field images. The
frequencies of the co- and contra-rotating orbits at the innermost stable
circular orbit (ISCO) in this geometry are also computed, in the hope that
quasi-periodic oscillations may permit to compare this model with Kerr's black
hole on observational grounds.
