The symmetries in the distribution of major volcanic centers on Mars have been analyzed by employing a technique involving the angular self-similarity of point fields. The distribution of all central vent volcanoes exhibits an axisymmetry along with a reflecting symmetry about a plane whose normal is the axis of axisymmetry. The latter axis is skewed by about 30 degrees from the rotational axis and is suggested to represent the early axis of symmetry of the interior dynamics. The high level of symmetry of Mars' volcanic distribution may be the result of a focusing of seismic energy from a major impact in the Hellas area that formed the Hellas basin and may have triggered the onset of volcanic activity in the greater Tharsis area. If the highland paterae near the Hellas region are not taken into account in the analysis, an asymmetry with respect to the reflecting plane appears. This is probably representative of the Late Hesperian and Amazonian, when the once global volcanic activity had retreated to the Tharsis province. These angular symmetries suggest a simple pattern of long-wavelength axisymmetric martian mantle convection involving a large stationary plume, as shown by our three-dimensional (3-D) spherical-shell simulations. On the basis of these calculations, we venture to propose that the volcanic history of Mars reflects the transformation of a bipolar mantle plume structure into a single megaplume by the process of plume-plume collision acting in concert with the deep Martian phase transitions. This transformation might have overcome the original pattern induced by a major impact forming the Hellas basin. We have repeated the procedure for the Earth's hotspot distribution, which revealed both axisymmetry and a certain level of reflecting symmetry. This similarity between the two planets is striking. The symmetries of the Earth's hotspot distribution are probably due to the interaction of the lower mantle megaplumes with the endothermic perovskite to gamma-spinel phase transition.