This paper explores the gauging of finite Abelian modulated symmetries in 1+1 dimensions, focusing on their dual symmetries and non-invertible reflection symmetries. Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way, generalizing concepts like dipole symmetries. The study uses local Hamiltonians of spin chains to analyze dual symmetries and their spatial modulations. It establishes sufficient conditions for the existence of an isomorphism between modulated symmetries and their duals, naturally implemented by lattice reflections. For prime qudits, translation invariance guarantees this isomorphism, while for non-prime qudits, techniques from ring theory show that the isomorphism can also exist, though not guaranteed by lattice translation symmetry alone. The paper identifies new Kramers-Wannier dualities and constructs related non-invertible reflection symmetry operators using sequential quantum circuits. These symmetries exist even when the system lacks ordinary reflection symmetry. The results are illustrated using various simple toy models. The paper also discusses the implications of these isomorphisms, relating them to new Kramers-Wannier dualities and non-invertible reflection operators. It presents a generalized Ising model for modulated symmetries, showing that its bond algebra always takes a specific form, allowing for gauging using Gauss's law. The paper concludes with an analysis of the non-invertible reflection operator and its implications for symmetry-protected topological phases and spontaneous symmetry breaking.This paper explores the gauging of finite Abelian modulated symmetries in 1+1 dimensions, focusing on their dual symmetries and non-invertible reflection symmetries. Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way, generalizing concepts like dipole symmetries. The study uses local Hamiltonians of spin chains to analyze dual symmetries and their spatial modulations. It establishes sufficient conditions for the existence of an isomorphism between modulated symmetries and their duals, naturally implemented by lattice reflections. For prime qudits, translation invariance guarantees this isomorphism, while for non-prime qudits, techniques from ring theory show that the isomorphism can also exist, though not guaranteed by lattice translation symmetry alone. The paper identifies new Kramers-Wannier dualities and constructs related non-invertible reflection symmetry operators using sequential quantum circuits. These symmetries exist even when the system lacks ordinary reflection symmetry. The results are illustrated using various simple toy models. The paper also discusses the implications of these isomorphisms, relating them to new Kramers-Wannier dualities and non-invertible reflection operators. It presents a generalized Ising model for modulated symmetries, showing that its bond algebra always takes a specific form, allowing for gauging using Gauss's law. The paper concludes with an analysis of the non-invertible reflection operator and its implications for symmetry-protected topological phases and spontaneous symmetry breaking.