  Modulated symmetries are internal symmetries that act in a non-uniform,
spatially modulated way and are generalizations of, for example, dipole
symmetries. In this paper, we systematically study the gauging of finite
Abelian modulated symmetries in ${1+1}$ dimensions. Working with local
Hamiltonians of spin chains, we explore the dual symmetries after gauging and
their potential new spatial modulations. We establish sufficient conditions for
the existence of an isomorphism between the modulated symmetries and their
dual, naturally implemented by lattice reflections. For instance, in systems of
prime qudits, translation invariance guarantees this isomorphism. For non-prime
qudits, we show using techniques from ring theory that this isomorphism can
also exist, although it is not guaranteed by lattice translation symmetry
alone. From this isomorphism, we identify new Kramers-Wannier dualities and
construct related non-invertible reflection symmetry operators using sequential
quantum circuits. Notably, this non-invertible reflection symmetry exists even
when the system lacks ordinary reflection symmetry. Throughout the paper, we
illustrate these results using various simple toy models.
