Ladybird riddle

If a ladybird is originally staying on a ${\color{#D61F06}{\text{red}}}$ cell, then it must fly into a ${\color{#20A900}{\text{green}}}$ cell.

If a ladybird is originally staying on a ${\color{#20A900}{\text{green}}}$ cell, then it must fly into a ${\color{#D61F06}{\text{red}}}$ cell.

There are $13$ $\space$ ${\color{#D61F06}{\text{red}}}$ and $12\space {\color{#20A900}{\text{green}}}$ cells, so it is not possible, that after the ladybirds have moved, there will be only ladybird in each cell.

It is not possible if you have an odd number of squares. In this case, you can see that all the bugs can move clockwise (or counter-clockwise), except for the one in the central square, which must then double-up in an adjacent square.

There is an easier way to intuitively see the answer to this problem by visually splitting the big square into smaller 2x2 or 2x1 squares and moving all bugs in those squares either clockwise/counter clockwise or up/down. By splitting the bigger square into smaller ones so that a circle or half circle(swap zone) can be drawn inside of each them we could see that eventually we will end up with a single spot where a bug have nowhere to go.

I did it the odd way, every move is a vector and the sum of all moves is 0 and it is impossible that 25 perpendicular vectors cancels out.

Every time a ladybug goes left, another one has to go right or vice versa. The same accounts for a ladybug going upwards or downwards.

Since there is an odd number of ladybugs, we can never find a solution with an equal number going right and left, and an equal number going up and down. We can however find a partial balance (horizontally or vertically), but the other direction will always fail to be balanced.

Let all ladybugs that can move in one direction (let's say, to the upward) do so. This would work for a group of 4 rows by 5 columns, or 5 rows by 4 columns. The problem is that whatever row or column is on the end, can't go anywhere, because they have to move to an adjacent square. So as a partial solution, we know that a mass migration of a large group of adjacent ladybugs on this scale, can't be accomplished. The next step I would get to from there would be, is there a smaller group, like a square of 4x4, that could all move in the same direction?