I guess I feel a little differently. To me, the thought and time (and maybe a little financial sacrifice) that go into shopping for and preparing a gift are part of the gift itself. You're not just giving someone a physical object, you're spending time choosing it for them and you pay for it with some of your "life energy" (to use the somewhat woo-woo term from Your Money or Your Life).
So even if the recipient doesn't know you re-gifted, I as the giver would feel a bit guilty because I only gave half a gift - I didn't devote the thought and time and money that would have gone into a "real" gift.
My siblings and I use Amazon wish lists to shop for each other at the holidays, and while I'm perfectly happy to receive things I put on my list, the gifts that mean the most to me are the surprise ones (not on my list) my sister chooses for me because she sees something she thinks I'll like, buys it, wraps it herself, and takes the time to put it in a box and mail it to me. That means a lot because there's a lot more to the gift than just the object.
So I don't regift. If I receive something I don't want, I give it to Goodwill or some other charity.