Mr. Hooper's reaction to seeing his own reflection is much the same as everyone else's reaction to seeing him wear the veil.
[...] his own antipathy to the veil was known to be so great, that he never willingly passed before a mirror, nor stooped to drink at a still fountain, lest, in its peaceful bosom, he should be affrighted by himself.
Mr. Hooper fears to see the veil, just as the members of his congregation do, and though Hawthorne never directly tells us why, we can make some assumptions based on, among other places, the end of the story.
By the story's close, Mr. Hooper is on his deathbed, tended by the Reverend Mr. Clark, who wishes to remove Hooper's black veil now that he is about to pass to the next life. Mr. Clark says, "'Suffer us to be gladdened by your triumphant aspect as you go to your reward. Before the veil of eternity be lifted, let me cast aside this black veil from your face!'" But Mr. Hooper adamantly refuses, and so Mr. Clark desires to know what great sin goes with Mr. Hooper to the hereafter. Mr. Hooper's final words are the most illuminating:
"Tremble [...] at each other! Have men avoided me, and women shown no pity, and children screamed and fled, only for my black veil? What, but the mystery which it obscurely typifies, has made this piece of crape so awful? When the friend shows his inmost heart to his friend; the lover to his best beloved; when man does not vainly shrink from the eye of his Creator, loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die! I look around me, and, lo! on every visage a Black Veil!"
In other words, Mr. Hooper has only ever worn a physical symbol of humankind's spiritual state. Just as the black veil has ever separated him from his fellows, so does the veil each of us holds up between ourselves and everyone else when we do not share our secret sins with each other. This is why Mr. Hooper tells his community to tremble at each other, not him; they are all sinful in nature -- it is not only he alone who carries the weight of secret sin. We each try to act as though we are sinless -- though we are all sinners (a popular Hawthorne theme) -- and in that pretending, we lie to one another, preventing ourselves from truly being known by or truly knowing anyone else. This lack of knowledge leads to a lack of understanding, and so we each perpetuate the myth of our sinlessness, and so others believe that they must do so as well. No one wants to feel as though they, alone, are a terrible sinner. And so we deny this part of ourselves, isolating ourselves from our fellows as long as we live.
Therefore, whenever Mr. Hooper would catch his own reflection, he would be reminded of this terrible state of humankind, reminded of his own sinful nature and the sinful natures of all those around him who insist on living a lie. The sight of the black veil over his own face would only be a reminder that even he was not brave enough to cast aside the figurative veil with which he has, as we all have, separated himself from everyone else. Thus, the veil is a reminder that he is not only sinful but also, in his own eyes, deceptive and cowardly.