I had the same thing happen to me about 8-9 years ago. It's why they don't let me handle sharp instruments any more! (Just kidding!) I was rotary cutting late one Sunday night & sliced off the tip of my left index finger. It bled like crazy. I went to work on Monday & it was still bleeding. I work with nurses but when my supervisor saw all the blood, she sent me to the ER. I still joke today that if you're bleeding to death or dying, don't wait for anyone at work to help you. Anyway, they couldn't reattach the severed piece of flesh (don't remember why). I have learned to keep my fingers out of the way.
I never heard of cutting gloves, though, so I will have to look into this.