I have noticed that for all the matrices representing quantum gates, if we read rows left-to-right and top to bottom, the read the same as columns top to bottom left to right.
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1& 0
(CNOT gate), if we read the values in from the first row, we see
1 0 0 0. Similarly if we read left-most column top to bottom. Second row reads
0 1 0 0, just like 2nd left column etc.
I believe this is a property of all matrices representing quantum gates, but cannot explain why exactly this is the case (and if it is, in fact, guaranteed for every quantum gate?)