The 2D Z₂ toric code admits a global symmetry exchanging electric and magnetic quasiparticles, known as electromagnetic duality. Known realizations include lattice translation symmetry, an exact Z₄ symmetry generated by a Clifford circuit, and an exact Z₂ symmetry generated by a non-Clifford circuit. We show that a Clifford electromagnetic duality cannot realize an exact internal Z₂ symmetry. This is proved rigorously for symmetries with coarse translation invariance by l lattice units for generic odd l. Therefore an exact internal Z₂ electromagnetic duality must be non-Clifford, whereas generic internal Clifford realization necessarily has Z_(2ᵐ) algebra with m≥ 2. Our result suggests an unexpected connection between exact electromagnetic duality and Clifford hierarchy of circuits.