"I thought the start of the tornado season was identical to the start of the hurricane season on the Pacific coast."

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It isn't. "Official" East Pacific hurricane season starts 15th May (1st June for the Central Pacific). For tornadoes the "season" is widely recognized as being March through June - the period when tornadoes most frequently occur, although there have been outbreaks in every month of the year. There are far fewer hurricanes outside of their designated season.

Stephen.