If \(u\) and \(v\) are two functions whose level sets have compact support, one can horizontally shift the support apart and define the “new coupled rearrangement” as Steiner symmetrization \((u+v)^*\) of \(u+v\). This is the geometric idea behind the new rearrangement in the title, and the main result is the inequality \(\int|\nabla(u+v)^*|^2<\int|\nabla u|^2+\int|\nabla v|^2\). (For functions with compact support this result can also be found in [B. Kawohl, Arch. Ration. Mech. Anal. 94, 227–243 (1986; Zbl 0603.49030)]. This strict inequality is then used to derive \(H^1\)-precompactness for minimizing sequences in a class of \(L^2\)-constrained minmization problems. Like many others in this field the author misspells Schwarz symmetrization (named after the geometer Hermann Amandus Schwarz) as Schwartz-symmetrization.