Statistical characteristics of freely decaying two-dimensional hydrodynamic turbulence at high Reynolds numbers are numerically studied. In particular, numerical experiments (with resolution up to 8192 × 8192) provide a Kraichnan-type turbulence spectrum Ek ~ k-3. By means of spatial filtration, it is found that the main contribution to the spectrum comes from sharp vorticity gradients in the form of quasi-shocks. Such quasi-singularities are responsible for a strong angular dependence of the spectrum owing to well-localized (in terms of the angle) jets with minor and/or large overlapping. In each jet, the spectrum decreases as k-3. The behavior of the third-order structure function accurately agrees with the Kraichnan direct cascade concept corresponding to a constant enstrophy flux. It is shown that the power law exponents ξn for higher structure functions grow with n more slowly than the linear dependence, thus indicating turbulence intermittency.