A common view regarding even-order nonlinear optical (NLO) effects, such as second-harmonic generation and the bulk photovoltaic effect, is that the breaking of inversion symmetry is necessary. However, this view relies on the assumption that light interacts with materials solely through its electric field. Here, we demonstrate that interference between the electric and magnetic components of the same light field can induce even-order NLO effects even in centrosymmetric materials. Using charge and spin photocurrents as prototypical examples, we present a unified microscopic theory showing how such electric-magnetic dipole interference (EMDI) drives NLO responses in inversion-symmetric crystals. Bilinear in both electric and magnetic fields, these responses scale linearly with light intensity. Combining symmetry analysis with first-principles calculations, we decompose the EMDI-induced photocurrents into shift-like and injection-like contributions and classify their dependence on crystal symmetry, time-reversal symmetry, and light polarization. Notably, we reveal that these EMDI-induced responses in centrosymmetric materials are weaker than conventional responses in non-centrosymmetric materials roughly by a factor of fine structure constant α ≈ [10⁻³--10⁻²] . Nevertheless, they remain well within experimental reach. Our work thus unlocks NLO functionalities in centrosymmetric materials for e.g., materials characterization, photonics and optoelectronics. This is vital given that many functionally importantly materials, such as cubic Si and cubic SrTiO₃, possess inversion symmetry.