Unlike other relics the relic of the True Cross is supposed to be kept veiled when it isn't exposed for veneration. One is supposed to genuflect to it when it is exposed. The priest is supposed to carry it either veiled or with a humeral veil.
The idea is the relic is Eucharistic. If it is the the Cross then it is drenched in the blood of Christ and treated as Blessed Sacrament, the Rite of Exposition is similar to Benediction.
In the light of the great debate: may the divorced and remarried venerate it?