I know there are things like the rule of thirds, the golden rule, the triangles, symmetry, but it's all based on one thing, Emphasis and Weight.
You don't even have to do a drawing based on a grid, because when you understand how to emphasis something, you can do a lot of good compositions without using a grid for it!

Obviously I can't give you a degree with my explanation but here's what I can fit in.

Emphasis (example, a bright orange dot in a grey and black background)

Balance (person fighting one corner of the piece, another person in the top corner jumping up and coming down on the opponent)

Proportions (comparing objects or people with other objects to determine the size of one another)

Repetition (a box with the same design elements each face, repeating design shapes on a character, etc)

Movement/Rhythm (lines moving towards your emphasis, example, an anime character going NANI?? with lines towards their face all around. Can be swirls and triangles and spirals, can be clouds and wind and leaves!)

Variety (interesting design elements that are different, but working together. Good for design)

Harmony (EVERYTHING added into one, but goes together!)

Everything above can be described with value, colour, size. I might do more parts, depending on how this topic goes with everyone's interest on this. Ciao!