Deliberate mistake: After Ramsey shows Sansa her flayed northern friend, when it cuts to Davos riding the horse through the encampment one of the shots is flipped. Note his sword and his cloak's crossed straps have switched to opposite sides.
Deliberate mistake: During the "previously on Game of Thrones" snippets, before the opening credits sequence, there are two flipped shots of Sansa and Jaime as she tells him that she has always wanted to be there for Cersei's execution, but she won't get the chance. (00:01:45)
Answer: As Hand of the King, Tywin is the second most authoritative person in Westeros behind only the king. The Hand's duties include sitting on The Iron Throne and dispensing order, justice and judgment, similar to what Ned Stark did in the Season 1 episode "A Golden Crown" while then-king Robert Baratheon was out hunting. It's also worth pointing out that Tywin's enormous wealth makes him the de facto most powerful person in Westeros, which he demonstrates in the Season 3 episode "Mhysa" when he sends King Joffrey to bed without his supper and remarks to Tyrion "You're a fool if you believe he's the most powerful man in Westeros. Do you really think the crown gives you power?"
Phaneron ★