Why don't cats look in the mirror?

Yes, but they think it's another cat and not themselves

ANSWER

when my cat was young it was a thing of great entertainment and mystery to him, but once he reached a certain age he would just look in the mirror as i held it to his face, (t was also entertainment to us too) and he'd look very bored and as if hes saying "yeah yeah yeah i get it , its a mirror, its me, how interesting"

They can see their reflection but since they don't smell "another" cat, they quickly lose interest. Cat society is governed by their sense of smell and body language. Without one, the other doesn't matter. In other Words, the cat will see the reflection, think it's another cat there but then, once it realizes there is no odor attached to it, it realizes it's not another cat.

My cat is vain, she looks at her self while cleaning, and sees where she missed and fixes it. She will stare at herself for 15 minutes or so and watch as she cleans.