Okay, gunna give this a shot after a request. Not sure how much I can actually teach here, so I'm going to base this whole thing about the things that go through my head as I'm shooting, which leads us firstly to what I like to call "Symmetry Play."
Symmetry has always been something that my eye has picked up on quickly. I think it's a good exercise in breaking down the elements of a photo and arranging them thoughtfully and also a good way to simplify your compositions. Symmetry is an easy way to break a composition down into neatly divided sections that you can use to play off of each other.
Now, in my view when I'm taking pictures of this sort... where my eye was caught because of some element of symmetry.. the image is going to be somewhere on a scale between completely symmetrical and totally asymmetrical. Now, I would venture to say that in photography, reaching complete symmetry can be tough to do (because the real world is rarely totally symmetrical) and because it can be a little boring.
A nearly totally symmetrical composition would be like this.
Pretty much every major element in the photo has mirror symmetry across the frame. Personally I find this type of vertical symmetry to be the easiest, but other types of symmetry are perfectly acceptable. This type of shot requires finding the perfect subject, and also the setting around the subject. Then you have to focus on the other smaller elements of the shot that are sneaking into the corners and what-have-you. Usually it's easiest to just move the camera to crop them out, but can you do this without compromising the overall composition?
Symmetry is usually easier to find in closeup, so you may want to look at the smaller things around you. Also, working close-up with objects is one of the best(easiest) ways to simplify a composition and give things a different perspective than the everyday view we're used to. If you're having a hard time finding total symmetry, look closer! Try to focus your composition down to smaller things that you can easily simplify around.
Now, the next idea is a little more fuzzy, and I'm calling it "Broken Symmetry" Now, I'm pretty sure that's an art school term. I never went to art school, so if I'm bastardizing it's traditional meaning, write me an angry email.
In my mind, this means just about anything that is in between full symmetry and full asymmetry. Typically this is going to be something that has elements of symmetry and asymmetry worked into the same composition. Can sometimes be harder than true symmetry, but I think it gives the composition a much more interesting tension and movement.
Nearly symmetrical, but there are some subtle things that throw it out of whack.
This one has much nearer to equal parts of symmetry and asymmetry.
So the assignment is just for 2 shots!
One to demonstrate near-total symmetry throughout the whole composition and one that uses elements of symmetry and asymmetry, otherwise known in my world as "Broken Symmetry"
Focus on keeping your compositions simple and free of extraneous elements that aren't contributing. Once you find your interesting symmetry in the world, simplify around it and only photograph the parts that interest you visually the most.
Go!