It's easy to understand, when you turn the brackets into readable characters.
o for opening bracket (
c for closing bracket )

So ()() would be ococ, which is coco backwards and clearly not a palindrome.

())( would be occo, which is a palindrome.

The brackets example is the one I use to drill into people's heads that it has nothing to do with visual symmetry

Meanwhile in my brain:

))<>((

thanks, satan.