  Exotic compact objects, horizonless spacetimes with reflective properties,
have intriguingly been suggested by some quantum-gravity models as alternatives
to classical black-hole spacetimes. A remarkable feature of spinning
horizonless compact objects with reflective boundary conditions is the
existence of a {\it discrete} set of critical surface radii,
$\{r_{\text{c}}({\bar a};n)\}^{n=\infty}_{n=1}$, which can support spatially
regular static ({\it marginally-stable}) scalar field configurations (here
${\bar a}\equiv J/M^2$ is the dimensionless angular momentum of the exotic
compact object). Interestingly, the outermost critical radius
$r^{\text{max}}_{\text{c}}\equiv \text{max}_n\{r_{\text{c}}({\bar a};n)\}$
marks the boundary between stable and unstable exotic compact objects: spinning
objects whose reflecting surfaces are situated in the region
$r_{\text{c}}>r^{\text{max}}_{\text{c}}({\bar a})$ are stable, whereas spinning
objects whose reflecting surfaces are situated in the region
$r_{\text{c}}<r^{\text{max}}_{\text{c}}({\bar a})$ are superradiantly unstable
to scalar perturbation modes. In the present paper we use analytical techniques
in order to explore the physical properties of the critical (marginally-stable)
spinning exotic compact objects. In particular, we derive a remarkably compact
{\it analytical} formula for the discrete spectrum
$\{r^{\text{max}}_{\text{c}}({\bar a})\}$ of critical radii which characterize
the marginally-stable exotic compact objects. We explicitly demonstrate that
the analytically derived resonance spectrum agrees remarkably well with
numerical results that recently appeared in the physics literature.
