We challenge the classical belief that products of degenerate quadratic forms must remain degenerate and we show that this fails in general, e.g. over tensor triangulated categories with duality. This opens new ways of constructing non-degenerate quadratic forms and hence classes in Witt groups. In addition, we encapsulate in a Leibniz-type formula the behavior of the product with respect to the symmetric cone construction. We illustrate these ideas by computing the total Witt group of regular projective spaces.