  Bernstein's inequality is a central result in the theory of $D$-modules on
smooth varieties. While Bernstein's inequality fails for rings of differential
operators on general singularities, recent work of \`{A}lvarez Montaner,
Hern\'andez, Jeffries, N\'u\~nez-Betancourt, Teixeira, and Witt establishes
Bernstein's inequality for invariants of finite groups in characteristic zero
and certain other mild singularities in positive characteristic. Motivated by
extending this result to new classes of singular rings, we introduce a
``two-sided'' analogue of the Bernstein-Sato polynomial which we call the
sandwich Bernstein-Sato polynomial. We apply this notion to give an effective
criterion to verify Bernstein's inequality, and apply this to show that
Bernstein's inequality holds for the coordinate ring of $\mathbb{P}^a \times
\mathbb{P}^b$ via the Segre embedding. We also establish a number of examples
and basic results on sandwich Bernstein-Sato polynomials.
