The anomalous behavior of liquid water is apparent from the temperature dependence of many experimental properties. Among these are the abnormal high. limiting ionic conductivities lambda(0)) of protons and hydroxyl ions, which in a previous paper we found to depend linearly on the square root of the absolute temperature T-1/2. Here we describe a further study of these conductivities in both light and heavy water, together with the remarkable transition temperature T-0 [242.7 K in H2O and 250.5 K in D2O], where the supercooled liquid becomes inert and, for example, restricted water molecule rotation is arrested. From T-0 the rotation barrier heights for the two solvents are determined. The conductivity data enable to obtain experimentally, the zero point energies of H+ and D+ in liquid water, with results in the right order of magnitude as compared to values estimated along quantum mechanical routes. Contrary to general opinion, the isotope effect lambda(0)(H)+/lambda(0)(D)+ is temperature dependent, its value being close to 2(1/2) only near 20 degrees C. The isotope effect lambda(0)(OH)-/lambda(0)(OD)- also takes temperature dependent values, substantially higher than 2(1/2). Still, the linear relationships with T-1/2 sustain a model based on water rotation control for the conductivity mechanism. For a quantitative analysis, the rotation frequency omega is expressed by a simple linear function of T-1/2 in terms of the moment of inertia I and the quantity T-0(1/2). This "ab initio" calculation is found to be in perfect agreement with theoretical and experimental data in the literature, when omega is identified with the reciprocal of the so-called single molecule relaxation time tau((sic)). The data analysis eventually leads to the conclusion that the hydrogen ion transfer between water molecules proceeds via two parallel pathways. Next to the rotation controlled hopping mechanism there exists a temperature independent transfer controlled by tunnelling, occurring at all relative orientations of the participating water molecules. The applicability of the excess conductivity concept for hydrogen and hydroxyl ions is critically discussed.