In all the art forms, there are rules for success. Whether writing, painting, or photography, you look like a rookie if you do not follow the rules. So when can you throw the rules away? I could make this a one paragraph article by simply saying "don’t!" But we can explore this.
In commercial fiction, for instance, each genre has some formulaic rules that have come about because the market dictates what is good and popular. It is all about selling books. When confronted with these "rules" the new writer proclaims "Well, Ernest Hemingway doesn’t write that way!" And what was your last name again? The swashbuckling pirates of old would refer to an old hand that had been at sea half their life as "salty." And it was probably more than just figurative speech. But there was also an old adage that proclaimed the wisdom "you can’t be salty until you are good." Once you have commanded the respect of your audience, then you can deviate from the norm and still be accepted, but not before.
These are a couple of examples. Depending on what your purpose is, such as stock photography, there may be many more, like leaving space for the title, text, advertising, etc.
So the first question is why would we ignore the rules? The rules are an indicator to the discerning, educated observer that we know the craft. And these rules are time proven to provide an attractive and enjoyable composition that conveys a message. The rules help you make "purty pitchers". Why deviate?
Before I state when you might toss the rules, let me throw in this caveat. Don’t use this explanation to throw out the rules. If you want to become a photographer that is recognized for doing good work, then most of your photographs need to follow the rules. Going back to the pirate reference from earlier which basically states that you must understand the rules to break them effectively. There are a few times when we would like to deviate from the rules of photography in order to make a statement, create artwork...or maybe the rules for some reason don't apply in that particular situation.
What are those times? When the deviation contributes to the theme such that it overrides the detraction of a rules violation, then it can be permissible. I say that with a slight pause, because you have to pull it off in the eyes of the beholder. I would use the terms novelty and shock value. Does the novelty of centering up your subject have value from symmetry that overrides the rule of thirds? You are trying to convey an image of a hasty retreat out of the picture such that you can have a merge that adds understanding to the message. These are times when perhaps it is permissible to deviate.
There are times when the rules of good exposure will be violated rather than those of good composition. You may have taken a picture in broad daylight, but want the ambiance of a twilight scene. Under exposure can get the job done. Or we dodge to underexpose and reduce the obviously busy background while our subject is a normal exposure. For graphic design, the monochromatic image is popular. A pasty blonde subject slightly overexposed on a dull background has some novelty and purpose, particularly if the subject is not really the model, but perhaps a piece of clothing or jewelry.
So in summary, can we disobey the rules? Yes, but beware. There needs to be an overriding purpose...something that communicates "I meant to do that", and it needs to really work before you toss out the standards for good photography. So add this to your checklist critique of your own images: would offbeat changes in composition, exposure, focus make this better? And if so, why? Of course, photography is a creative art. So we are going to be creative. Just remember there is a fine line between genius and insanity. Don’t step over it too often!