A motivation for the development of atomically resolved spectroscopic imaging STM (SISTM) has been to study the broken symmetries in the electronic structure of cuprate high temperature superconductors. Both the d-wave superconducting (dSC) and the pseudogap (PG) phases of underdoped cuprates exhibit two distinct classes of electronic states when studied using SI-STM. The class consists of the dispersive Bogoliubov quasiparticles of a homogeneous d-wave superconductor. These signature are detected below a lower energy scale |E| = D0 and only upon a momentum space (k-space) arc which terminates near the lines connecting k = ±(p/a0, 0) to k = ±(0.,p/a0). This 'nodal' arc shrinks continuously with decreasing hole density. In both the dSC and PG phases, the only broken symmetries detected in the |E|≤D0 states are those of a d-wave superconductor. The second class of states occurs at energies near the pseudogap energy scale |E| .D1 can be associated with the 'antinodal' states near k = ±(p/a0,0) and k = ±(0.,p/a0). These states break the expected 90°-rotational (C4) symmetry of electronic structure within CuO2 unit cells, at least down to 180°-rotational (C2) symmetry (nematic) but in a spatially disordered fashion. This intra-unit-cell C4 symmetry breaking coexists at |E| .D1 with incommensurate conductance modulations locally breaking both rotational and translational symmetries (smectic). Empirically, the characteristic wavevector Q of the latter is determined by the k-space points where Bogoliubov quasiparticle interference terminates and therefore evolves continuously with doping. The properties of these two classes of |E| .D1 states are indistinguishable in the dSC and PG phases. To explain these two regimes of k-space that are distinguished by the symmetries of their electronic states and their energy scales |E| D1 and |E|≤D0, and to understand their relationship to the electronic phase diagram and the mechanism of high- Tc superconductivity, represent key challenges for cuprate studies. © Vladimir Dobrosavljevic, Nandini Trivedi & James M. Valles, Jr., 2012. All rights reserved.