The Ten no Ryu are the organisation that fight for mankind. They do not wish to see humanity be destroyed by the Chi no Ryu, so have vowed to fight them. All fights take place within a kekkai that the Ten no Ryu create, as the Chi no Ryu don't feel that they need to protect mankind from their fights.
An intruiging part of the manga is that every single member of the Ten no Ryu has a job outside of the organisation.