It is said that a maiden handed her handkerchief to our Lord on His way to Calvary. He wiped the sweat from his brow, returned the handkerchief to the owner, and went on. The handkerchief was found to bear a perfect likeness of the Saviour, and was called Vera-Iconica (true likeness), and the maiden was ever after called St. Veronica. One of these handkerchiefs is preserved at St. Peter's church in Rome, and another in Milan cathedral.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894