There are very few things in nature that are truly symmetrical. Plants don't grow straight up towards the sky with leaves sticking out symmetrically all over the place; they grow efficiently, towards the light, which is not always up, and their leaf growth pattern matches the direction. Even a redwood or pine tree, though it grows straight up, is not symmetrical. Its branches don't start until part way up, where they reach wide with fewer leaves where there is less light. At the top, the branches become shorter, with as little shade as possible on all the leaves up there. It's a waste of energy to grow equally in all directions at once.
Human nervous systems are the same way. They find balance in asymmetry. It takes a huge amount of energy (neurologically speaking) to teach both sides of your body exactly the same skills. Just like how a tree grows more leaves where it can get sunlight, the human nervous system uses its energy efficiently, sending energy all over the body for gross motor skills (walking, running, jumping, etc), but focusing energy for finer motor skills on one side (writing, brushing your teeth, kicking a soccer ball, etc).
Symmetry is an attractive and powerful myth. It's simple, pleasing, and accessible. I teach my clients to learn to pay attention and notice how a movement feels on one side of their body, and new client after new client expects a given movement to feel and behave the same on the other side, but they are consistently surprised. The reality check is this:
No one has a symmetrical body.
Ever seen one of those photos of someone's face photoshopped to be truly symmetrical? It's kind of creepy. Asymmetry actually creates interest and a different kind of beauty, speaking in design and art terms.
No matter how hard you try to make your system symmetrical, everyone has a dominant hand, a dominant leg, a dominant eye, even - get this - a dominant side of your mouth and throat. The way that you learned to move as a tiny baby started that dominance. Let's say the door to your nursery was to the right of your crib, and your parents always lay you down such that you could see the door with your right eye. Every time you heard the door open, you looked at it to see who was there, making your right eye stronger and more dominant with each repetition. As you grew up a little and started to walk, the balance you developed formed around that right eye dominance. You still learned how to use your left eye, but in a different way than your right. You can do all the eye strength training you want later in life, and that right eye dominance will always be there. If natural selection didn't like a little asymmetry, evolution would have a taken a very different path than it has.
So, next time you find yourself wondering or worrying about why carrying your bag feels so much easier on one side than the other, or why your feet are slightly different sizes, don't worry. Some amount of asymmetry is normal.