Whether recirculating or reciprocating, exercises may be bilaterally (left/right) symmetrical, but aren't necessarily so. Exercises that are not bilaterally symmetrical should be performed on both sides (both mirror-image versions) to approximately the same degree.
There are other symmetries, which can usually be described in terms of rotation, either circular movements or the twisting of a hand, foot, shoulder, or hip to its limit. Circular movements are generally reversible and should usually be performed in both directions. Twisting may also be reversed, although doing so usually causes the original motion to become awkward and in need of other modifications if it is to continue to flow smoothly.
Gently oscillating between opposite twists is a fine, and a good way to familiarize yourself with the limits of movement, but doing so can quickly become boring. It helps to combine joints, for instance rotating one hand as far as you can, twisting shoulder, forearm, and wrist all in the same direction, perhaps twisting the other hand at the same time, either in the same direction or in the opposite direction.
By combining arm/hand and leg/foot twisting together, you will discover that there are stress lines, like rubber bands pulled tight, which you could trace through muscles and tendons, extending continuously from toe to fingertip. These are not the meridians of Chinese medicine, but a chain of tissues that are all being stretched at the same time. You'll also find that the second step in reversing the twisting motion (after relaxing the muscles that put you into a twist in the first place) is to contract all of the muscles along that chain, and that if you continue to do so the result will be an approximately opposite twist.
These long lines, composed of chains of muscle and tendon, when uniformly stretched and developed, are capable of generating movement that is both graceful and powerful, because it involves the entire body.