I have a theory (now I'm serious) about Saya's behavior before the tragedy, where she wasn't acting like herself. Maybe that was because of Yukianesa's influence?
I'm seeing it like that: Saya found Yukianesa somewhere in church (if there was remains of Black Beast, why there can't be a Nox?), or maybe somewhere in the forest (maybe Terumi left it there, knowing that she'll pick it up later), and she fell under it's influence. Later, to get rid of sword that messed with her head, she gave it to the brother who's been bullying her - Jin.