Is it permissible for Muslims to celebrate Eid al-Adha with Christians?

Eid al-Adha is one of the rituals of Islam, and it is inconceivable for Christians to participate with Muslims in its celebration in the manner of the Eid prayer and takbir, due to the exclusivity of this for the people of Islam. However, there is no harm in their participating by offering congratulations and showing joy to Muslims. Nevertheless, their congratulating us does not permit us to congratulate them on their holidays, because their congratulating us is a congratulation on truth, while our congratulating them is a congratulation on falsehood.