the face he looks at each day and has no clue what’s inside it

[Coffee Break]

I encountered the expression "the face he looks at each day and has no clue what’s inside it", but am struggling to understand it. Could you please let me know what it means? Here is the excerpt:

“I am so tired of him. He’s the healthiest man in the world, and I’m the worst thing for him. There are days when, I swear, all I want is to seize the pumice stone in my bathroom and bash my face in with it, because it reminds me of the face he looks at each day and has no clue what’s inside it, no clue, no clue. He made me stop being who I am; worse yet, I stopped knowing who I was.”

- André Aciman, Eight White Nights, First Night

This is a novel published in the United States of America in 2010. It is narrated by a nameless male protagonist. He meets Clara at a Christmas party in Manhattan. Now Clara is speaking about her ex-boyfriend Inky, as to how she is tired of him.

I wonder what the underlined expression means.
I first thought that Inky ("he") is looking at the face of Clara, but then it seemed to me that Clara is exasperated at Inky's face because she had no clue about his face...

So I wonder "the face he looks at" means "the face Inky gazes at", or "the face Inky shows".

And I also wonder whether the subject of "has" in "has no clue what's inside it" is "he (=Inky)".

 

[Tarheel]

He knows her, but he doesn't know her. He doesn't know the real person (the person inside). Apparently, she wishes he did know her better.

 

[5jj]

There are days when, I swear, all I want is to seize the pumice stone in my bathroom and bash my face in with it, because it reminds me of the face he looks at each day and has no clue what’s inside it, no clue, no clue.

... all I (Clara) want is to seize the pumice stone in my (Clara's) bathroom and bash my (Clara's) face in with it (the pumice stone), because it (my (Clara's) face) reminds me (Clara) of the face he (Inky) looks at each day and has no clue what’s inside it (my (Clara's) face) , no clue, no clue.

 

[Coffee Break]

@Tarheel and @5jj,

Thank you very much for the explanations.
Oh, so she is saying how Inky, her ex-boyfriend, does not understand what is behind her (Clara's) face!
Inky looks at Clara's face each day, but has no clue as to what is inside Clara's face. He does not understand her mind behind the face.
So "no clue, no clue" would perhaps mean Clara's exasperation at Inky's ignorance regarding her emotions that occur behind her face.

Probably the use of "the face" rather than "my face" confused me. But now I think I grasped it all thanks to you!
I truly appreciate your help.

 

[Tarheel]

Everybody has speech habits -- you, me, Clara. The more we get to know somebody the more we get used to their speech habits. As for why Inky doesn't know her any better than he does, that's mostly Clara's fault, isn't it?

 

[5jj]

Probably the use of "the face" rather than "my face" confused me.

It's the face he looks at each day. My face would be incorrect.

 

[Coffee Break]

@Tarheel and @5jj,

Thank you very much for the explanations.
Indeed, I agree that Clara has a very unique speech habit!

It's the face he looks at each day. My face would be incorrect.

Probably, I guess, the combination of "he looks at + my face" would be grammatically incorrect; the combination of "he looks at + the face" would be correct, though I am not sure...
I truly appreciate your help.

 

[5jj]

Probably, I guess, the combination of "he looks at + my face" would be grammatically incorrect; the combination of "he looks at + the face" would be correct, though I am not sure..

Both are possible grammatically.

Only the is possible in It reminds me of the face he looks at each day.