Why is it illegal to play with a left handed hockey stick?

Answer 1

In Field Hockey, the use of different-handed sticks creates a danger scenario where two players of opposing teams wil end up colliding when both go for the ball, because their bodies are on the same side. This is a major safety problem, and is dealt with by simply removing the possiblity of it ever occurring.

In Ice Hockey, there is no such rule. Sticks are not particularly right- or left-handed, and players may use them on either side - some ambidextrous players might use it on both.