There is a school of thought that interprets magical devices in fairy tales, not as depicting magic, but as a kind of futuristic technology. One might think the same about magical mirrors. But although the analogy is certainly suggestive, I think it leads us in the wrong direction.
There is a certain kind of personal gain which you can only attain by deliberately becoming invisible to others — by ensuring that they can no longer see you for what you really are. If this sounds like a small price to pay, wait to see what develops out of it: for there are hidden costs to the bargain. It is, after all, a pact with dark forces, and in the end, it may leave you in despair.
The adventure is kicked into motion by an episode of emotional upheaval (and we know where that leads): a surprise meeting with Julia, an old flame, at the New Year’s party. Romantic memories awake, then confusion arises when she acts somewhat coldly but also, it seems, with some recognition, and finally deep disappointment fuels the mix at her husband’s appearance. This is too much: it all...
Mirrors — those of the symbolic flavor, i.e. mirrors of the soul — don’t necessarily have to be visual. In one of Neil Gaiman’s short stories, a narrative work of art (i.e. a story-in-a-story) does the same trick that Oscar Wilde’s painting of Dorian Gray performs. Thus Gaiman removes the symbolic mirror one step further from literal mirrors.
Imagine you sit model for a painting. When it is finished, you look at it with surprise and admiration: the painter is a true artist, and this is his masterpiece. It brings out something about you, your true personality; it shows things of which you were only dimly aware yourself. To use an old expression: it reflects your soul.