Antisymmetry of real quadratic singular moduli

We confirm a conjecture of Darmon-Vonk on the antisymmetry of real quadratic singular moduli. The proof relies on a careful analysis of rigid meromorphic cocycles à la Darmon-Gehrmann-Lipnowski for the split orthogonal group on four variables. Moreover, we prove the modularity of a generating series of Kudla-Millson divisors in the spirit of Gross-Kohnen-Zagier.