During the first two phases, the DPS legendary ring effect should be used to help kill the Doomfire Spirits and Hellfire Deathcallers, as they are high-priority targets. During Phase 3, depending on how rings work when phased, players in the Nether Realm could use the legendary ring to quickly kill the Shadowed Netherwalker and escape, while the rest of the raid could use their ring to deal additional damage to Archimonde and the Infernal Doombringers.
The healing legendary ring effect should be used to heal up after heavy raid damage, such as the Shadowfel Burst during Phase 1, Unleashed Torment in Phases 2 and 3, and Rain of Chaos or Hellfire in Phase 3.
The tanking legendary ring effect should be used to mitigate the heavy damage of Death Brand in Phases 1 and 2. It is unclear how these rings will work when one tank is phased into the Twisting Nether.