Their relationship was a secret, but it was never his secret alone.
He holds to that as he stands in self-imposed exile, reduced to a mere spectator while Cordelia lays the coil of her hair on the fire. There are strands of his hair inside that coil, strands she cut last night in the tense silence following their argument about where he would stand today. Now her offering hides his so both can burn together in front of everyone.
It's so very like her, to give him what he won't claim for himself. To abandon the argument without abandoning him.