Joseph Braun, SJ posited that around the 9th century deacons wore the chasuble during stational processions, but removed the vestment when entering the church. During penitential times they retained the chasuble rather than don their ornate officials' uniforms. The "folded" part of the folded chasuble shares an origin with the practice of lifting the priest's vesture during the incensations and elevations (not that they originated at the same time, they just originated because of the same problem). Chasubles were normally made of layered wool, often embroidered and layered with additional materials to create motifs, scenes, or ornamentation; some even carried heavy jewels and ivory veneers. The ample cuts that prevailed until the Reformation broke out combined with the heavy properties made movement very difficult. Since deacons and subdeacons did quite a bit of carrying things (lectionaries, the gifts, the chalice) they began to roll up the fronts of their vestments.
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Sadly, the Western way of adapting the priest's vestments to his need to move was an inelegant one: cut off the sides. Silk, which does not breath as well as light wool or cottons, became the preferred material for vestments during the Renaissance and the age of exploration. At first they merely reduced the scope of the vestment, as with the "Borromean" and "Philippian" chasuble styles. Eventually, someone just cut off the arms and snipped the thing at the knees to create the sandwich board style we all know and don't love. It would be wrong to call the vestments most popular before Vatican II, and hence what it almost exclusively used in Traditionalist circles today, "Roman." Roman vestments were a bit more ample, almost like the Borromean and Philippian styles, and had the perpendicular inserts in the back, a continuation of the medieval parish chasuble style. The dominant pre-Vatican II style is really French: still material, bulging maniples, open fronts, no break in the back, and always a Cross on the back. As with most baroquerie, it can work, but only with the best and most expensive of materials. Otherwise, the chasuble becomes a canvass for kitsch.
In a fit of ludicrous regulation befitting an Italian bureaucracy, the Vatican decreed that traditionally cut Latin rite vestments should not be made without the permission of Congregation for Rites. Pius XI eventually dismissed this rule, but it ranks with some of the sillier things anyone has seen worthy of regulation.