Fetching details from the dictionary...

face; mug
surface
cheek meat
彼は怒ったつらで睨んできた。
He glared at me with an angry face.
朝起きたら寝不足のつらだった。
When I woke up, I had the face of someone who didn't sleep enough.
あいつのつらは見たくもない。
I don't even want to see that guy's face.