I had the "blue and green" rhyme quoted at me when I was a child. As I always was a contrary thing (nothing has changed!) I used to say "what about bluebells then, they look nice and they have blue flowers and green leaves?". Got into trouble for answering back! But I still think that certain shades of blue and green look lovely together.
Red and green are at opposite sides of the colour spectrum. Artists use this fact to good effect in landscapes - Constable and Corot, for instance, would put a tiny amount of red (a person in a red jacket, or a red roof) in a predominantly green landscape. You might not even notice it at first, but cover it up and you can see it does make a difference.
And at the opticians, some of the tests have red and green backgrounds.