WRT what is symetrical, or not, it depends upon the perspective. From the perspective of the arrow, the string is all it "sees." And if you bend a string together at the nock point you'll see more dramatic asym in the so called "even" bow where arrow pass is arbitrarily established at 2" above dimensiional center. And for no other reason or justification that I can find than because somebody suggested it. In fact in antiquity, where ideas are more thoroughly evaluated, arrow pass closer to center is very well represented.
Couple of points where I differ. First, when upper limb is farther from the string, that is called positive tiller. This is the most common config, even if arrow pass is at center. It's simply a matter of degree, when arrow pass is closer to center less (positive) tiller is needed. But due to the effect of the geometry of the string hand being necessarily above the bow hand (can't tshoot through ones own hand), some degree of positive or even ideally keeps the nock travel up then down instead of vice versa. Folks generally find nock travel in the opposite direction not as easy to tune for, ie. requiring higher nock point. Second, the lower limb is rarely if ever under more strain. It is stiffer yes, but doesn't bend as far as it is longer. Strain is all about bending radius. It may store more energy, and certainly delivers more, relatively, by typically it enjoys a better string angle.
Arrow pass at center bows have the apex of the curve at the arrow pass. So called symetrical bows, more accurately shorter upper limb bows, apex is at dimensional center of handle, or under ones palm on the bow hand. As such one must adjust their eye for proper tiller, incorporating the handle in the lower limb rather than ignoring it, and/or how we setup on the tiller tree.
Regardless of the config, balance is imperaitve. Geometry won't combat one limb taking more set, or hinging. That's all about the process of the crafting, not so much design.
In practice I find a happy medium w/ arrow pass about 3/4" to 1" above center. What suits you best may be different, based largely upon how you place pressure with your bow hand and string hand. One test I use is, I want relatively less positive tiller, and relatively less nock height, about an 1/8" being as close to the edge as I care to tread.