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The discovery of high-temperature superconductivity in the cuprates more than three decades ago triggered a tremendous amount of scientific activity, yet it has remained a challenge to understand the unusual ‘normal’ state from which the superconductivity evolves upon cooling. I will review our collaborative neutron scattering, X-ray scattering, torque magnetometry and charge transport results for HgBa₂CuO_4+δ, arguably one of the most ideal cuprate compounds for experimental study due to its high structural symmetry and high optimal superconducting transition temperature of nearly 100 K. From the comparison with data for other cuprates we are able to separate underlying universal properties from compound-specific behavior. Our results point to the existence of two electronic subsystems, and to an essential role played by nanoscale inhomogeneity inherent to these lamellar oxides. We conclude that the combination of inhomogeneity and two-component physics drives percolative behavior on multiple energy scales, giving rise to universal characteristics of the normal state and of the emergence of superconductivity.