Omagatoki: Adversity grants all allies an additional damage multiplier based on the percentage of HP the individual attacker has lost. The formula is most likely 1*(1+%HP LOST), so the multiplier will be at around x2 when the user's HP is nearly depleted. The attack also boosts the power of recovery skills, likely in the same fashion.
The effect lasts until the end of the faction's turn. The Magatsuhi gauge needs to be full and used up to execute the skill. As it's a Magatsuhi skill, no icons are used, and the turn isn't passed to the next demon.