Abstract: We discuss constructions and (non)existence of invariant bilinear operators on (almost) Hermitean symmetric spaces. In particularly, based on ideas of invariant quantization, we show how linear invariant operators can obstruct existence in the bilinear case. This turns out to be crucial in the study of symmetries of the Laplacian and related operators. The curved setting is more involved. Nevertheless, we present certain general class of (curved) pairing of solutions of first BGG operators and also suggest how to deal with curved symmetries of the Laplacian.