I promise, this is not beating a dead horse.
Yes, yes, I know, triggering their reactions on the defense is a bad idea. Because it means the attacker - y'know, your opponent - gets to resolve the ring twice. But talking things over with my Scorpion buddy, @Facedownninja , we realized that the ring effects are unconditional. So are the Reactions. So, if you trigger the reaction, then the ring effect must be resolved a second time. How could we use this to our advantage? For all of these "I" am the defender, and "you" are the attacker.
Ring of Void - Let's say I only have one character with fate, and only one fate on them. If you swing with Ring of Void, you'll resolve it to take the fate off my character. But then I make you resolve it again, forcing you to remove a fate from one of your own characters.
Ring of Earth - You've been playing pretty heavy on the card draw, or it's late in the game and your deck is getting low. If you swing with Earth, I can force you to draw a second card, potentially and eventually forcing a mill.
Ring of Air - This one's dangerous, but might have potential. You're Scorpion. You like to be under me, but still keeping it close. If you swing with Air to try to even out the honor a bit, I could force you to hit it again. Hopefully, this would put your honor above mine, shutting down some of your more powerful cards.
I know they're all highly circumstantial, and pretty counter-intuitive. And I'm not saying it's the best idea in the world. But under certain specific circumstances, it could be beneficial to force your opponent into resolving rings twice.