Within the French tradition, two forms of cross-shaped orphrey patterns can be found which have come to be distinctly associated with two important sees of France: Lyon and Paris. Lyon is, of course, home the "Primate of the Gauls" (the Archbishop of Lyon) and has its own particular liturgical rite -- or it may be more appropriate to now call it a "use" given the process of Romanization that took place in its regard since Trent.
In the case of Lyon the orphrey design is made up of a single column orphrey with the horizontal bars of the arms of the cross simply attached to that orphrey. In the case of Paris, we have a Latin cross shape whose axis has been expanded in the centre, rather like inserting a square in the middle of the join of the cross. The following illustration, taken from a book explaining the rites and ceremonies of the Lyonese liturgical tradition, displays the difference -- and you will note they go so far as to point out that the Parisian form is forbidden in Lyon (no doubt in part to preserve their own unique and distinctive liturgical identity and tradition).
How these specific designs developed is an open question, though it would seem in the case of the Lyonese form that it may well have been a case of a more typically Italo-Roman form (which has the single column orphrey on the back) simply having had the horizontal arms added to it, thereby forming a cross. The rite of Lyon was frequently characterized by its generally more conservative approach in preserving and using antique liturgical forms so it may well be the case that the so-called Lyonese form simply represents a form of orphrey that was at one time used far more widely throughout France.
Whatever the case, here are a few more examples coming from these two traditions as they came to develop.
