Integration-by-parts reduction of scalar operators (${\varphi }^2(\partial \varphi {)}^2\simeq -{\varphi }^3\square \varphi /3$)

Two Lagrangian densities differing by a total derivative yield the same action under suitable boundary conditions. For example ${\partial }_{\mu }({\varphi }^3{\partial }^{\mu }\varphi )=3{\varphi }^2(\partial \varphi {)}^2+{\varphi }^3\square \varphi$. Likewise $\varphi {\square }^2\varphi \simeq -\partial \varphi \cdot \partial \square \varphi \simeq (\square \varphi {)}^2\simeq ({\partial }_{\mu }{\partial }_{\nu }\varphi {)}^2$. These are algebraic relations modulo boundary terms, not uses of the equation of motion.