A fun, but not very reliable weather saying tells us:
If March comes in like a lion it will go out like a lamb. It often happens that March begins cold and stormy and ends warm and balmy. People tend to think that nature, and life, always strives for a balance, which may have been why the adage was originally created.
Another theory about the origin and meaning of this saying is that it’s astrological: At the beginning of March, the constellation Leo (the lion) is on the eastern horizon at sunset. By the end of the month, Aries (the ram) is on the western horizon. †
Shirley Byers’ book “Never Sell Your Hen on a Rainy Day” explores over 100 weather rhymes and sayings. It is available from McNally Robinson at: http://www.mcnallyrobinson.com/home.