The women in Roy's life are a strange lot. First there's Harriet Bird, who tries to murder him. Then there's Memo, who turns out to be a gold-digging backstabber. And finally, there's Iris, whose role isn't clear at first. The novel usually sticks a white flower on these ladies. Let's try and figure out what the deal is with that.
The first blossom shows up with Harriet, when she gets on the train.
She had dropped a flower. Roy thought it was a gardenia but it turned out to be a white rose she had worn pinned to her dress. (1.62)
A gardenia can mean secret love, and a white rose can symbolize innocence. So when Roy decides to "get rid of" the white rose that he picked up and put in his pocket, that might tell us something about him losing his innocence. And, in fact, Harriet Bird is the woman who will take away his innocence with a silver bullet.
Later on the white flower shows up on Iris (who happens to share a name with a flower…hm!):
a young blackhaired woman, wearing a red dress, was sitting at an aisle seat in short left. He could clearly see the white flower she wore pinned on her bosom [. . .] (6.105)
A scene like that has gotta freak Roy out, after his trauma with Harriet. The fact that Iris wears a white rose makes us think that Roy's afraid of women in general, and that the white rose is, for him, a sort of attractive but dangerous sign. He wants them, but he also runs from them.