Rows and columns in quantum-gate matrices read the same - why?
A counter-example to your observation is the Pauli Y gate:

$$Y = \begin{bmatrix} 0&amp;-i\\i&amp;0\end{bmatrix}$$

The first row is 0 then -i, but the first column is 0 then +i.

Another example is the X->Y->Z->X gate (a 120 degree rotation around the X+Y+Z axis of the bloch sphere):

$$\frac{1-i}{2} \begin{bmatrix} 1&amp;-i\\1&amp;i\end{bmatrix}$$



The reason the rows often look so much like the columns, particularly for self-inverse operations, is because quantum gates are unitary and the inverse of a unitary matrix is its conjugate transpose. But you can easily break the symmetry by multiplying an individual row (or column) by a phase factor, which produces a different but valid quantum operation.