  Let $X$ be a complex projective algebraic variety with Gorenstein quotient
singularities and $\X$ a smooth Deligne-Mumford stack having $X$ as its coarse
moduli space. We show that the CSM class $c^{SM}(X)$ coincides with the
pushforward to $X$ of the total Chern class $c(T_{I\X})$ of the inertia stack
$I\X$. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever
is defined, coincides with the pushforward to $X$ of the total Chern class
$c(T_{II\X})$ of the double inertia stack $II\X$. Some consequences concerning
stringy/orbifold Hodge numbers are deduced.
