  We determine the extent to which the collection of $\Gamma$-Euler-Satake
characteristics classify closed 2-orbifolds. In particular, we show that the
closed, connected, effective, orientable 2-orbifolds are classified by the
collection of $\Gamma$-Euler-Satake characteristics corresponding to free or
free abelian $\Gamma$ and are not classified by those corresponding to any
finite collection of finitely generated discrete groups. Similarly, we show
that such a classification is not possible for non-orientable 2-orbifolds and
any collection of $\Gamma$, nor for noneffective 2-orbifolds. As a corollary,
we generate families of orbifolds with the same $\Gamma$-Euler-Satake
characteristics in arbitrary dimensions for any finite collection of $\Gamma$;
this is used to demonstrate that the $\Gamma$-Euler-Satake characteristics each
constitute new invariants of orbifolds.
