Anticyclonic storms usually form around high-pressure systems. These do not "contradict" the Coriolis effect; it predicts such anticyclonic flow about high-pressure regions. Anticyclonic storms, as high-pressure systems, usually accompany cold weather and are frequently a factor in large snowstorms. Jupiter'sGreat Red Spot is a well-known extraterrestrial example of an anticyclonic system.
Anticyclonic tornadoes often occur; while tornadoes' vortices are low-pressure regions, this occurs because tornadoes occur on a small enough scale such that the Coriolis effect is negligible.