When is it correct to salute with your left hand?

Not real sure

You lost your right arm and still have your left.

When your right had is occupied or in a sling.

If your arm is injured. I had one incident where my salute and greeting was questioned by an officer. I addressed a group of six 2nd LT's with "Good morning gentlemen!", they all returned the salute but a conversation ensued afterward. They took my word for it on the reg. but I could imagine them dressing down the next Marine that didn't greet them the same way. :))

I would have to say never

When carrying the colors

I believe it's only when your right hand is injured.

When your right hand/arm is either a) broken or in a cast; or b) full of papers/equipment/gear essential to the job at hand.

If your right arm is injured and incapable of rendering a proper salute

Navy/ Marine Corps permits a left handed salute when you are a guidon bearer who is behind the formation or if you have a right hand/ arm injury that prevents you from saluting properly.
A guidon bearer who is in front of the formation is required to render a salute by tilting the guidon forward until it is parallel to the deck.