Spoiler for true identity of the twins:
Shion was actually born as Mion, and Mion was born as Shion.
Since "Mion" (that is, the Shion that we know) is the rightful heir to the Sonozaki family, she was favored over "Shion" (the Mion that we know) ever since they were little.
"Mion" and "Shion" disliked the different treatments they got: they're the same person, so why should one be treated different from the other?
That's why they began swapping identities so they could both enjoy the privileges that "Mion" has. Sometimes Mion would become Shion, and Shion would become Mion. In a way, both of them are Mion, and both of them are Shion.
However, one day, while "Shion" was pretending to be Mion, she was taken away and was given the demon tattoo which marked her as the next head of the Sonozaki family.
... Starting from that day, "Shion" lost her identity as Shion, and became Mion. "Mion" lost her identity as Mion, and became Shion.
That's why Mion has always felt bad towards Shion for stealing her identity, because it was Shion (the original Mion) who should've been chosen as the next head of the Sonozaki family, not her.
When Shion locked Mion up inside the underground jail cell, they finally reverted back to their original identities, hence the "onee" from Mion.
You can also see Shion referring to Mion as "Shion" in the final TIPs as well.
Now you know the true meanings behind Mion and Shion's coversation inside the library!
That's pretty cool to know although I'm not sure if I'm more confused now or less confused. In the end up Meakashi-hen when "Mion" is leading K1 down to the torture chamber she says "You know, Shion really liked you." then seconds later she says "Mion liked you too." Why is she referring to herself in the third person in both cases?
Still, later in that Ep when "Mion" has K1 in the torture chamber K1 says stuff to the extent that "Mion is will always be my best friend. The Mion inside me will never die." etc. I see now why "Mion" is moved to tears by that.