I believe cats recognize themselves in a mirror.
We got our cat as a feral cat that simply moved into our house while it was under construction. Although not a kitten she was quite young and not fully grown. We eventually decided to keep her, took her to the vet for a checkup, shots, spading, etc., then brought her to our apartment to await the completion of our house. (note: This "free" cat cost us over $300 in vet bills!)
When she got to the apartment she explored the place and came across a mirror. Apparently she had never seen one before. Her tail fluffed out, her back arched, she let out a big hiss and took a swipe at the mirror. She thought it was another cat! A few moments later, after looking behind the mirror and coming at it from different angles, sniffing, poking, etc., she ignored it. She no longer shows any interest in herself in the mirror.
I'm pretty sure she figured out it was her in the glass. She certainly recognized it as a cat and shortly thereafter realized it wasn't *another* cat. Perhaps she figured it was just a very good picture of a cat, but she also recognizes cats as cats on TV (shows extra interest) but doesn't freak out.
I have tested her understanding of a mirror. If she has a mirror in her field of view and I am in the right spot I can wave my hands and she will turn to look at me.
I've never read anything about feline cognition, I'm a dog guy. But, the reading I've done says that the test for self-recognition is to put them in front of a mirror with a small dab of odd colored paint on their forehead, or a ssomething otherwise abnormal to their coat. If they try to remove it, it's self-recognition and at least in the case of canines, fairly rare.