  In strongly correlated electron materials, the electronic, spin, and charge
degrees of freedom are closely intertwined. This often leads to the
stabilization of emergent orders that are highly sensitive to external physical
stimuli promising opportunities for technological applications. In perovskite
ruthenates, this sensitivity manifests in dramatic changes of the physical
properties with subtle structural details of the RuO$_6$ octahedra, stabilizing
enigmatic correlated ground states, from a hotly debated superconducting state
via electronic nematicity and metamagnetic quantum criticality to
ferromagnetism. Here, it is demonstrated that the rotation of the RuO$_6$
octahedra in the surface layer of Sr$_2$RuO$_4$ generates new emergent orders
not observed in the bulk material. Through atomic-scale spectroscopic
characterization of the low-energy electronic states, four van Hove
singularities are identified in the vicinity of the Fermi energy. The
singularities can be directly linked to intertwined nematic and checkerboard
charge order. Tuning of one of these van Hove singularities by magnetic field
is demonstrated, suggesting that the surface layer undergoes a Lifshitz
transition at a magnetic field of ~32T. The results establish the surface layer
of Sr$_2$RuO$_4$ as an exciting 2D correlated electron system and highlight the
opportunities for engineering the low-energy electronic states in these
systems.
