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.
Note to self: Religion - freaky
Thanks for correcting my faulty memory. Point was, I found it kind of freaky that they kept track of stuff like this, especially the breast milk of the Virgin Mary (according to my friend the Vatican was not too happy about that one.)
We did see quite a few first order relics, fingers, bones, etc of various Saints in the various cathedrals from Notre Dame to the ones in Rome.
We stayed at the Santa Ana hotel, very near St. Peters. According to my friend, we could put all our souvenirs from the various local shops on the window sill, within the sound of the Pope's voice, and they would be 'blessed' by the Pope (JPII at the time). I had bought myself several pairs of the bright red socks at the vestment shops, hoping my feet would be infallible. Still can't dance.