The question was how do you start lunging with a caveson (initially present it), not how, or the progression (which was recently discussed in another thread).Didnt actually put why. And tinyliny is correct about the tradition (was specifying that...not the op...sorry).
A cavesson is not just a glorified halter for a number of reasons, they are padded, have straps (not a throatlatch) to prevent them from sliding sideways and torqued into the eye of the horse. And because the line is attached in front rather than underneath, so they allow vibrational half halts, easier flexion inside which can makes a horse more upright (with s.r.) and without can allows stretching more easily. But in a halter with line attached underneath usually cause the horses response to any vibration from underneath by going higher. When a halter is attached to one ring it simply pulls the neck inside and the horse counter bends/leans.
Cavesons were initially used (a couple of hundred years ago) for work around a pillar, and for lateral flexibility (with draw reins attached with a rider). A caveson DOES teach a horse about working into a connection (and how it was progressively presented), and self carriage (vibrate the line). THen when the sr are attached to a bit it is not a strange/frightening concept to be worked into connection. It is progressive, the horse knows HOW to respond to hh, will stay up and open and less likely to have the bars impacted aggressively. It is a STEP in training.
Lunging is to present ideas progressive without the weight of the rider, this include contact, balance, change of gaits, lateral flexibility, steady tempo, etc. By first using the s.r. to the caveson and later to the bit that is allowed, methodically.
And initially the s.r. may be even, but usually the inside is slightly shorter (1-2") to allow the beginning of lateral flexibility (Reiner Klimke, Egon von Neindorff, SRS does this amoung others). It underlies the idea of riding 'in position'/seeing the inside eyelashes and working into the outside reins. It is using what the horse needs, and there are some where the outside is shorter if their behavior/movement needs it. Because the horse should flex/bend lightly through the body on a circle (to increase engagement/flexibility lateral) oftentimes the inside sr will get too loose, connection is missing (more likely to happen in a bit that onto caveson), hence slightly shorter inside. Always remember this point of working on a curved line with side reins is to develop lateral flexibility/balance/being upright which helps with straightness, it is NOT to enforce longitudinal flexion or lowering in the neck.