Flipping rows and columns of a board

Idea. Find a quantity that no move can change.

Invariant. Fix two rows $r\ne r'$ and two columns $c\ne c'$, and consider the four squares $(r,c),(r,c'),(r',c),(r',c')$. Look at the parity of the number of black squares among these four.
- A move on row $r$ (or $r'$) changes exactly two of them.
- A move on column $c$ (or $c'$) likewise changes exactly two.
- A move on any other row or column changes none of them.

Changing two squares changes their black count by $-2$, $0$ or $+2$: the parity of the black count is invariant.

Conclusion. At the start this number is $0$, so it stays even forever. Now suppose a board with a single black square $(a,b)$ could be reached. Choose any other row $r'\ne a$ and column $c'\ne b$ and the block of the rows $a,r'$ and columns $b,c'$: it contains exactly one black square, an odd number. Contradiction.

Remark. The reachable boards are exactly those in which every $2\times2$ block has an even number of black squares.

Lesson. Invariant: restrict attention to a small sub-configuration on which each move acts in a controlled way.