  We investigate the properties of the Cheeger sets of rotationally invariant,
bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariant
Cheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of pieces
of Delaunay surfaces, which are rotationally invariant surfaces of constant
mean curvature. We show that if $\Omega$ is convex, then the free boundary of
$C$ consists only of pieces of spheres and nodoids. This result remains valid
for nonconvex domains when the generating curve of $C$ is closed, convex, and
of class $\mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of the
fact that, for general nonconvex domains, pieces of unduloids or cylinders can
also appear in the free boundary of $C$.
