We have studied the atomic force microscopy (AFM), X-ray Bragg reflections, X-ray absorption spectra (XAS) of the Pd L-edge, Scanning electron microscopey (SEM) and Raman spectra, and direct magnetoelectric tensor of Pd-substituted lead titanate and lead zirconate-titanate. A primary aim is to determine the percentage of Pd⁺⁴ and Pd⁺² substitutional at the Ti-sites (we find that it is almost fully substitutional). The atomic force microscopy data uniquely reveal a surprise: both threefold vertical (polarized out-of-plane) and fourfold in-plane domain vertices. This is discussed in terms of the general rules for Voronoi patterns (Dirichlet tessellations) in two and three dimensions. At high pressures Raman soft modes are observed, as in pure lead titanate, and X-ray diffraction (XRD) indicates a nearly second-order displacive phase transition. However, two or three transitions are involved: First, there are anomalies in c/a ratio and Raman spectra at low pressures (P = 1 - 2 GPa); and second, the c/a ratio reaches unity at ca. P = 10 GPa, where a monoclinic (M_c) but metrically cubic transition occurs from the ambient tetragonal P4 mm structure in pure PbTiO₃; whereas the Raman lines (forbidden in the cubic phase) remain until ca. 17 GPa, where a monoclinic-cubic transition is known in lead titanate.