We define integer valued invariants of an orbifold Calabi-Yau threefold X with transverse ADE orbifold points. These invariants contain equivalent information to the Gromov-Witten invariants of X and are related by a Gopakumar-Vafa like formula which may be regarded as a universal multiple cover / degenerate contribution formula for orbifold Gromov-Witten invariants. We also give sheaf theoretic definitions of our invariants. As examples, we give formulas for our invariants in the case of a (local) orbifold K3 surface. These new formulas generalize the classical Yau-Zaslow and Katz-Klemm-Vafa formulas.