Monte Carlo and molecular dynamics computer simulations using the rigid TIP4P and the flexible BJH intermolecular H2O potentials were carried out for 50 states of supercritical water characterizing a very wide range of thermodynamic conditions, 573 < T < 1273 K; 0.02 < rho < 1.67 g/cm3; 10 < P < 10,000 MPa. Good agreement with available experimental data of the simulated thermodynamic and structural properties give confidence to the quantitative statistical analysis of intermolecular hydrogen bonding under the conditions studied. Energetic, geometric, and angular characteristics of supercritical H-bonds and their distributions at a given temperature remain almost invariant over the entire density range studied from dilute gas-like (~ 0.03 g/cm3) to highly compressed liquid-like (~ 1.5 g/cm3) states. The increase of temperature from ambient to supercritical affects the characteristics of H-bonding in water much more dramatically than the changes in density along any supercritical isotherm. Compared to H-bonds in liquid water under ambient conditions, the H-bonds at 773 K are, on average, 10% weaker, 5% longer, and less linear. Both above and below the H-bonding percolation threshold the fractions of H2O molecules involved in a certain number of H-bonds in liquid and supercritical water closely follow the universal binomial distribution as a function of the average number of H-bonds per one water molecule in the system, as predicted by the independent bond theory. This universal distribution remains intact even when dynamic criteria of H-bonding lifetimes are additionally applied.