As a student, I was told never to crop. Reason given: a strong artist knows how to design and cropping is a sign of weakness. It reduces the value of the painting. I choose to blow off this expert advice and crop whenever it will make a better painting.
What's "cropping"? It's the process of cutting down a painting's dimensions by reducing the horizontal or vertical sides or both. I've even converted a vertical painting into a horizontal and found more than one successful small painting in an unresolved large one.
I've had a fair amount of success cropping. The photos show how I use mat corners to find potential crops. I often find new takes on a successful painting that I wouldn't crop, but would start a new painting based on this technique.
As far as a reduction in value, I just don't believe it. Your job as an artist is to produce the best image possible. If the cropped version looks stronger, go for it!