Rate symmetry is involved with the regulation of a mechanical watch. It is connected to the fact that every cycle of the balance wheel consists of two swings, first in one direction and then in another. At every swing, the impulse pin on the balance roller shifts the pallets across, from one side to the other of the banking pins. So the balance wheel swings through a certain degree of arc from where the impulse pin moved the pallets to where the balance wheel stops and begins its return swing. If the arc of swing is not the same on both sides, the watch is said to be out of beat. Watch repairers can measure this using using an electronic timer machine, with differences between the two sides of the cycle of the order of a few milliseconds.
Some watches are fitted with a beat corrector, in addition to the regulator which adjusts the rate of the watch. The beat corrector is where the outside end of the hairspring terminates, and so rotating the beat corrector effectively moves the position of the jewel on the impulse pin. When the balance is at rest, the impulse jewel on the balance roller should correspond to a line running from the pallet pivot, through the banking pins and on to the balance pivot.
If the watch doesn’t have a beat corrector, the external end of the balance hairspring will be joined to a fixed stud, and can’t be adjusted. In this case, rate symmetry can be regulated by rotating the hairspring collet, which is slid onto the balance staff, and to which the internal end of the hairspring is attached. There is a notch in the brass collet that can be used to rotate it on the staff.
A watch such as the Glashütte Original PanoMaticInverse has two swan-neck regulators, one for the regulator itself, the other for rate symmetry.
For a clear video explanation of rate symmetry and beat correction, see the Watch Repair channel on YouTube.
Below, a photo of the balance wheel with the more customary style of beat corrector, courtesy Guy Sie/flickr.com