The only way I can see to change the right-hand spell (other than going into the Spell menu, but I'm asking about hotkeys here) is to press the hotkey twice, which first changes the left-hand spell, and then to press the hotkey for the left-hand spell again, which switches it back to what you had before. Is this the only way, i.e. going through the left hand to change the right hand? Or is there some other way?

Thanks.

and is it possible to do it the other way around for weapons?

I am not aware of any other way to do it using the only the hotkeys. You could bring up the Favorites menu and select the spell in either hand there. But it defeats the whole point of hotkeying it...

That's what I do as well, get spell and weapons in the hands you want them to be.

Then with the whatever weapon in my right hand I hotkey that situation, and hitting that hotkey I got both correctly I seem to remember.

You can alter the hand in Quick keys as well before you number hotkey your situation, I think, without checking.

In other words, using Hotkeys, you have to cycle through the hands. Not sure why it does not start out in the Right as that is the dominant hand. Perhaps it was chosen to go the left hand first from a Hotkey as people would generally keep their main melee weapon in their Right Hand, and switch out Spells with the Left. Problem is, you end up cycling the Spell or Weapon to the Right Hand to change the spell or weapon in the Left.