Is there such a thing as "secondary virginity"?
I spent my freshman year of high school at a private Protestant school. They taught that one could have secondary virginity if one asked. Is there anything similar in the Catholic Church? Besides confession, is there a way to be cleansed of sins against chastity in particular?
No, there is no such thing as "secondary virginity" because virginity is both a physical and a spiritual state of being. Once someone has willingly engaged in "the act," one cannot return to a state of never having done so. The only exception might be those cases in which there was no consent to the act. In that case, the bodily evidence of virginity may have been lost but the state of never having freely engaged in "the act" cannot be "stolen" against the will of the individual.
As for reparation after a sin of this kind, the sin is cleansed in confession, as are all sins that involve grave matter. Acts of reparation after sacramental confession for having offended against chastity should be discussed with the confessor or with a spiritual director.