Strongly correlated electron materials exhibit an intimate relation between charge, spin and structural degrees of freedom, leading to new emergent phases which seemingly break the symmetries of the underlying crystal and result in often unexpected sensitivity to external stimuli. The members of the Ruddlesden-Popper-series Sr_n+1Ru_nO_3n+1 exhibit a wide range of properties attributed to such physics, including unconventional superconductivity, metamagnetic quantum criticality and ferromagnetism. We show in a detailed study of the surface electronic structure of Sr₂RuO₄, an unconventional superconductor, how tiny structural distortions lead to a significant reconstruction of the Fermi surface and the low energy electronic structure. We use ultra-low temperature Scanning tunnelling microscopy to establish the existence of four van Hove singularities within 5 mV of E_f, as well as checkerboard charge order and nematicity of the electronic states. Including these orders in a tight-binding model gives excellent agreement with the experiment. By applying a magnetic field up to 14 T, we observe one of the van Hove singularities to Zeeman split, with one branch extrapolated to reach E_f at ~32 T – providing a text-book example of tuning towards a magnetic field-driven Lifshitz transition.