Pairing the twins and putting them in a room doesn't change the proportion of people who rolled a six. It just finds a six-roller for each non-six-roller and sits them down across from each other.

That you get put into a room isn't evidence about your roll. Whatever you rolled, you still get put into a room, and you get the exact same experience. What you learn by getting paired up is information about the joint state of you and your twin's rolls - without the pairing, there are 36 possibilities, and with the pairing there are 10 (which one of you rolled the six, and five ways for the other roll to turn out).