Ab initio molecular orbital and density functional theories have been used to study the structures and binding energies of the dimers of phosphinic acid (PA) and its dimethyl derivative (DMPA). For the first compound we have located all possible minima of the potential energy surface, while for the second only the most stable dimer was considered. The geometries were fully optimized at the MP2(full)/6-31+G(d,p) and B3LYP/6-31+G(d,p) levels of theory. The harmonic vibrational frequencies were evaluated at the same levels, while the final energies were obtained using a B3LYP/6-311+G(3 df,2p) approach. Both phosphinic acid and its dimethyl derivative form cyclic dimers in the gas phase, where the two monomers are held together by hydrogen bonds (HBs) which are significantly stronger than those found for their carboxylic analogs. The estimated dimerization enthalpies for PA (23.2) and DMPA (23.2 kcal/mol) are the highest reported so far for neutral homodimers in the gas phase and almost twice those measured for formic and acetic acid dimers. For the particular case of DMPA this estimated value is in very good agreement with the experimental one (23.9+/-6 kcal/mol). As a consequence of the large strength of the HBs, the activation barriers associated with the concerted double proton transfer are also sizably smaller than those predicted for their carboxylic analogs. These barriers become negligibly small when zero point energy (ZPE) corrections are taken into account and therefore tunneling must be very efficient. The calculated harmonic vibrational frequencies for the most stable DMPA dimer are consistent with its experimental infrared (IR) spectrum in the gas phase, which shows a characteristic ABC structure of the nu(OH) band, typically associated with strongly hydrogen bonded complexes. Internal cooperative effects are not negligible in the case of phosphinic acid dimers, and the P = O ... H hydrogen bonds in the global minimum are about 1.0 kcal/mol stronger than those found in other stable dimers where only one of these linkages exists.