Almost always. Under the specific ceremonial circumstance of piping a senior naval officer aboard ship, the Boatswain's Mate doing the piping, his right hand being occupied with his bosuns' whistle, will make do saluting said officer with his left. AFAIK, no one else anywhere goes to this kind of trouble.
The "rifle salute" if that's what it's called, is something I can't speak with authority on -- where your saluting hand goes to the stacking swivel of your rifle at Order Arms. Seems only to be done by sentries posted at the front door, and isn't much practiced nowadays.
Though I did find
this about rifle salutes on an Anzac Day page, last paragraph. I think the US military adheres to a similar standard.
A US manual from quite a few years back