Most of my fractals are symmetrical. Symmetry has an inherent appeal, because we are attracted to harmony, balance and order. Also, human beings and nearly all animals have symmetrical bodies, which must help form our notions of beauty.
Fractals are abstract and often chaotic-looking, so if the fractal is cropped to make it symmetrical then it becomes less strange and more digestible. Going a step further, I have made a fair few fractals with two axes of symmetry, by stitching together copies of the same fractal, as in this example.
These images are neat and orderly and some are rather effective, resembling Persian carpets or mandalas. If the elements of the image are complex and interesting then it can still look rich and exotic, even if it has two axes of symmetry.
However, the downside to having so much symmetry is that it can look boring, static and predictable. All art has to find a balance between repetition and novelty. Too much repetition makes the work boring and bland, whereas too much novelty makes it chaotic and disjointed.
So we seek a balance that is pleasing and one that looks "right" to our eyes. This is, of course, subjective. Every person has a different degree of liking for symmetry.