Alfred breaks the mirror because his smile reminds him of Ivan.
Not exactly, of course. It's a sweeter, dociler smile. It's a wide, white, everything-is-all-right smile, a of-course-I'm-not-worried-not-angry-not-disagreeing-not-anything smile. A housewife smile. But the falseness is there, just like in Ivan's. Falsely sweet.
It's not so much a smile he puts on when at meetings, an act put on for others. It's the smile he forces himself to wear when his paranoia is panting behind him like a large furry black beast, the smile finds himself wearing while he's cleaning the house. He cleans to keep his mind off things, to drive away the growing fear of Ivan, of communism, of himself. Oh, here's something red! A vase? Very nice, was it a gift from someone? He can't remember, but that's not important, is it? It's red. Throw it away.
It is the smile he wears when he says he is nothing like Ivan.
It is the smile he wears when he says he will never be like Ivan.
It is the smile he wears when he lies to himself.
And it takes a mirror for him to see that his smile is making him more like the person he fears and hates.
