Why is a hamming (7,4) coding scheme symmetric?

I'm trying to understand the explanation on pages 2/5 of the file and 18 in the book http://www.inference.org.uk/mackay/itprnn/ps/17.21.pdf on why the (7,4) Hamming code is symmetric. The explanation follows on from the solution of problem 1.6 which wanted is to find the block error $$p_B$$ and bit error $$p_b$$ of the (7,4) hamming code. I don't quite get what the explanation is getting that though?

The goal is to show that under any noise vector of weight 2 a source bit is as likely to be flipped as a parity bit, this is just stated but then the explanation goes on. Could someone explain what the section "Symmetry of the Hamming (7, 4) code" is getting at in relation to this? They switch the first and fifth vector and get equation (1.50).

A few questions i have

1. Do all code words t that satisfy Ht = 0 for H in 1.49 also satisfy H't=0 for H' in 1.50? If so what is the intuition here?
2. How does the explanation show symmetry in relation to problem 1.6? Why does it mean parity bits are as equally likley to be flipped as source bits?

I'm reading this book for fun and would really appreciate any help to allow me to understand this fully thanks!