Classifying closed 2-orbifolds with Euler characteristics

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.