power consumption in a XOR I'm trying to understand how a XOR affect traces of power consumption in a Differential Power Analysis on smartcards.

What's the difference in energy radiated in a $1 \oplus 1$ and in a $0 \oplus 1$?

If I understand correctly we have a XOR gate, and if a $1$ is xored with a $0$ then electrons in the 1 branch just continue flowing thus we still have some energy compared to a $0 \oplus 0$.
The exact details depend on the logic family but basically The difference is that the charge representing those "1" bits need to be discarded somehow. This is done by switching the output to ground, 1^0 is 1 so no charge need to go away but 1^1 is zero meaning at least two gates get discharged to ground slightly raising its voltage due to the non zero resistance in the ground return path.

Though, this really depends on the logic Family, the effect of charging the output gate lowering vcc may be dominant in some like you say. But in most any mos family, some assymetry will be noticed. ECL is notably immune due to steering currents that always flow not charging capacitors.