In this work, we investigate the one-dimensional $XX$ lattice model and its cousins through the lens of momentum and winding $U\left(1\right)$ symmetries. We distinguish two closely related ${\mathbb{Z}}_2$ symmetries based on their relation to the $U\left(1\right)$ symmetries, and establish a web of ${\mathbb{Z}}_2$-gauging relations among these models, rooted in two fundamental seeds: the $XY\pm YX$ models. These two seeds, each self-dual under gauging of the respective ${\mathbb{Z}}_2$ symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the $XX$ model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the $c=1$ compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.

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• Conformal field theory
• Quantum phase transitions
• Quantum many-body systems
• XY model