Pair rings are metal rings meant to be attached to a sword hilt. When a sword bearer first receives her sword and stars, she also receives a silver colored pair ring, which she may exchange with any other single swordbearer to become a shinyuu pair. The rings attached to their hilts signify that the two are a pair.
I know what you’re probably thinking, and yes, it is like tying the knot. In fact, Hayate and Jun love referring to husbands and wives, much to Ayana’s chagrin. People do pair up for reasons other than friendship, though. This is made obvious by the pairings of Claire and Minori, Sid and Nancy, and so on. It is possible that they pair up simply to be able to test their dueling skill, or to earn money. And some of these girls definitely need the money.
When two people split up, they switch out their silver pair rings for black ones, which signify that they are not only lone wolves, but ex-partners. Once a person pairs up again (if they ever do), they usually switch that black ring with their silver one again. However, some don’t. Mudou Ayana has chosen to attach her silver pair ring directly onto her black one, perhaps to remind herself of her old partnership.