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We discuss duality and mirror symmetry phenomena of Landau-Ginzburg orbifoldsconsidering their elliptic genera. Under the duality (or mirror) transformperformed by orbifoldizing the Landau-Ginzburg model via some discrete group ofthe superpotential we observe that the roles of the untwisted and twistedsectors are exchanged. As explicit evidence detailed orbifold data arepresented for $N=2$ minimal models, Arnold's exceptional singularities, $K3$surfaces constructed from Arnold's singularities and Fermat hypersurfaces. (Toappear in the proceedings of the workshop, ``Quantum Field Theory, IntegrableModels and Beyond'', Yukawa Institute for Theoretical Physics, KyotoUniversity, 14-18 February 1994.)Comment: 27(portrait)+3(landscape) pages, LaTeX, KEK-TH-409. (one reference  added

Duality of Orbifoldized Elliptic Genera