I'd add on that we don't know the nature of the of the possession itself. The seers themselves could be like Anders, fully in control of their own minds with a spirit of a certain virtue within them, offering advice or embodying that particular virtue for their society.
For all we know, (and there's no evidence to support this, just merely supposition), the Riviani could have a council of seers, each one possessed by differing virtues, Justice, Mercy, Valor, and so on, and when it comes to making decisions, each of the spirits give an opinion based on their extremities, and the political body can make their decisions based on what the Seers/Spirits suggest as solutions, maybe a combination of ideas.
Justice is swift, cold and hard, and is without mercy. Valor is about glory on the battlefield, so righteous fury and vengeance would work hand-in-hand there, but opposing virtue spirits may offer alternatives that the former spirits would never consider because it's not part of their nature.
Without more information, I cannot actually say how the Rivaini work. We only know that the seers allow themselves to be possessed, the non-mages violently opposed the Chantry as a cultural thing because the Chantry wanted to lock up all the seers in the Circle, and the book Asunder makes it clear that because some seers kept in contact with their families, a Seeker blabbed about it, and then the entire circle, down to the last man, woman and child, were killed. Not because of abominations, not because of blood magic, but because mages were in contact with their families.
That's all we truly know about the Rivaini.