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By old results with Millson, the generating series for the cohomology classes of special cycles on orthogonal Shimura varieties over a totally real field are Hilbert-Siegel modular forms. These forms arise via theta series. Using this result and the Siegel-Weil formula, we show that the products in the subring of cohomology generated by the special cycles are controlled by the Fourier coefficients of triple pullbacks of certain Siegel-Eisenstein series. As a consequence, there are comparison isomorphisms between special subrings for different Shimura varieties that may be of motivic origin.