Most lure sold - in fact all lures sold - are symmetrical in shape and action. Crankbaits, jig skirts & trailer, grubs, spoons or anything else are symmetrical.
Round lures in general have symmetrical halves on either side and most lures move side-to-side equally or spin in a circle. But is symmetry something fish need to provoke bites or is that mostly for anglers? Growing up buying and using symmetry-based lures for decades seems like the shape and action fish might expect seeing as how most of the animals they consume are. But as lure crafter and tester, I've found it not always necessary as long as a lure's action and size meet strike requirements.

Natural looking or acting is one thing anglers have in their heads as a requirement for the strike. But fish that strike lures don't know natural from unnatural but rather focus their senses on motion and speed once shape and size are sensed. In fact something about a lure that is asymmetrical may work if not as well as conventional shapes. Even symmetrical lures like a floating Rapala ripped & paused on the surface is not symmetrical in it's path and gets blasted by shallow bass.

Just an observation.

asymmetrical seems like would impair the retrieve. That would be one of the first obstacles to over come on hard baits.
Plastics like craws, and creature baits with asym appendages may not be a problem.

You're right - crankbaits must have symmetry to run right. The lures mentioned were examples of lure we take for granted must be symmetrical.
Soft plastics - not so especially many grub and some plastic worm tail designs like the Kut Tail.
Asymmetry examples would be the black and red prongs sticking out of these grubs:


Another would be the top stick wacky rigged:


The tail of the Kut Tail looks like a curl tail cut in half, but it's far different than curl tails or paddle tails.


Man I sure like using a candle to stick together different lure parts:


..... such as when making these asymmetric lures:

I see symmetry in those lures, depending on the axis.
Nature is symmetrical.

If you can reproduce a lure with a mold, it has symmetry.

I suppose, a Mobius strip doesn't have easily defined symmetry.

What about a mold with asymetrical halves? Just because there's a mold doesn't mean one side is a mirror image of the other, does it?

its flipping.