We report on recent extensions to the work (to be published) on PbHPO4 by one of us (RJN) with R. N. P. Choudhary and B. Březina. The structure of PbHPO4 in its paraelectric and ferroelectric phases has been investigated with high resolution (sin θ/λ > 1 Å−1) single crystal neutron-diffraction techniques. Data were collected at one temperature above TC (310 K) and five below, on the D8 diffractometer at the Institut Laue-Langevin, Grenoble. Structure refinements indicated that the phase transition is order-disorder for the hydrogen but displacive for the heavy atoms. Above TC the hydrogens are equally distributed (50:50) over two centrosymmetrically related sites in short “disordered” hydrogen bonds, and become fully ordered (100:0) on one of these sites at temperatures well below TC. This increasing ordering of the hydrogen with falling temperature follows, within error, the curve of spontaneous polarization, ps, versus temperature: the saturation of ps occurs over an unusually wide temperature range from TC to below TC - 100 K. But the displacements of the heavy atoms (approximately perpendicular to the hydrogen bond) are apparently already close to their saturated values only a few degrees below TC. This striking difference in behavior between the hydrogens and the heavy atoms- difficult to interpret in terms of possible crystal microstructures in the ordering range- has now been subjected to detailed testing. Constrained refinements and statistical significance tests on a number of models, including disorder models for the heavy atoms, now leave little room for doubt that the marked difference between the temperature dependence of the hydrogen ordering and of the heavy-atom displacements is a real one. Yet, as will be discussed, this conclusion remains difficult to reconcile with (i) conventional ideas about crystal microstructure; (ii) the notion of a transition “driven” by the hydrogen ordering, and (iii) the observed behavior of ps as a function of temperature.