Okay should I alter the seasons to fit the natural order, or leave them as is.

As is - Winter(Serpent), Summer(Phoenix), Spring (Dragon), Fall/Autumn(Tiger)

Natural Order according to seasons - Spring(Serpent), Summer(Phoenix), Fall/Autumn(Dragon), Winter(Tiger).

The way the Seasons relate to the elements - as Follows

Spring - Electricity(as in Thunderstorms)
Summer - Heat
Fall/Autumn - Wind.
Winter - Cold.