The Catholic church does a pretty good job of defiling itself, if you ask me (a Catholic).
On a trip to Italy a friend who lives in Rome explained Catholic relics to me. As I recall, a first order relic would be a piece of the true cross, his robe, the shroud of Turin, etc. A second order relic would be something associated with Mary. A church in Rome supposedly has some of Mary's breast milk in a reliquary.
Third order relics are objects that have been touched or in close proximity to first order relics. They tend to abound in Italy.
I'm surprised I read through the first 100 posts and found no mention of AA. They also did a pretty good number on Alcoholics Anonymous and there doesn't seem to be anyone complaining.
A first-order relic is an actual piece of a saint (or Christ, if one existed). A second-order relic is something the saint or Christ wore or used during his lifetime. A third-order relic is something that has touched a first- or second-order relic.
Thus, the cross should technically be a second-order relic, the Shroud of Turin third-order.