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The discovery of unconventional superconductivity in magic-angle twisted bilayer graphene (tBLG) supported the twist-angle-induced flat band structure predictions made a decade earlier. Numerous physical properties have since been linked to the interlayer twist angle using the flat band prediction as a guideline. However, some key observations like the nematic phase and striped charge order behind the superconductivity are missing in this initial model. Here we show that a thermodynamically stable large out-of-plane displacement, or corrugation of the bilayer, induced by the interlayer twist, demonstrates partially filled states of the flat band structure, accompanied by a broken symmetry, in the magic-angle regime and the presence of symmetry breaking associated with the superconductivity in tBLG. The distinction between low and high corrugation can also explain the observed evolution of the vibrational spectra of tBLG as a function of twist angle. Our observation that large out-of-plane deformation modes enable partial filling of states near the Fermi energy may lead to a strategy for offsetting the effects of disorder in the local twist angle, which suppresses unconventional superconductivity and correlated insulator behavior in magic-angle tBLG.