This paper systematically investigates the gauging of finite Abelian modulated symmetries in 1+1 dimensions, focusing on local Hamiltonians of spin chains. Modulated symmetries are internal symmetries that act non-uniformly and spatially. The authors explore the dual symmetries and their potential new spatial modulations after gauging. They establish conditions for an isomorphism between modulated symmetries and their duals, implemented by lattice reflections. For prime qudits, translation invariance guarantees this isomorphism, while for non-prime qudits, it can exist but is not guaranteed by lattice translation symmetry alone. Using this isomorphism, they identify new Kramers-Wannier dualities and construct non-invertible reflection symmetry operators using sequential quantum circuits. These non-invertible reflection symmetries exist even when the system lacks ordinary reflection symmetry. The paper includes various toy models to illustrate these results.This paper systematically investigates the gauging of finite Abelian modulated symmetries in 1+1 dimensions, focusing on local Hamiltonians of spin chains. Modulated symmetries are internal symmetries that act non-uniformly and spatially. The authors explore the dual symmetries and their potential new spatial modulations after gauging. They establish conditions for an isomorphism between modulated symmetries and their duals, implemented by lattice reflections. For prime qudits, translation invariance guarantees this isomorphism, while for non-prime qudits, it can exist but is not guaranteed by lattice translation symmetry alone. Using this isomorphism, they identify new Kramers-Wannier dualities and construct non-invertible reflection symmetry operators using sequential quantum circuits. These non-invertible reflection symmetries exist even when the system lacks ordinary reflection symmetry. The paper includes various toy models to illustrate these results.