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Here is Conway's notation for the complete classifications of all 2-orbifolds. The star means that the disk is removed and the boundary of the disk is silvered (so, it is just a mirror acting by reflections). All other symbols have the different meaning depending on if they go before the star or after.
Examples: is a cylinder; is a sphere; is a football with planar angle at each end; is an infinite pillow.
The Euler number can be nicely defined on orbifolds. Here is the practical way how to find an Euler number of a 2-orbifolds if you have the Conway notation for this orbifold.
Then the Euler number of an orbifold is 2 minus the sum of the corresponding values in the table, where we say that a symbol has a full value if it goes before the star and half value otherwise.
A good 2-orbifold has hyperbolic, euclidean or spherical structure iff its Euler number is respectively.