This study investigates the control of elastic wave propagation in one-dimensional periodic waveguides through twist-based geometric variations to alter the symmetry of the Kelvin cell. By imposing such twists, the original lattice topology is preserved, while mirror symmetries are selectively broken through a single geometric parameter, allowing the dispersion characteristics to be adjusted without requiring embedded resonators or substantial mass augmentation. Using a complex-valued Bloch–Floquet framework, we demonstrate that these twist-induced modifications activate two distinct attenuation regimes: traditional Bragg scattering and polarization-dependent band gaps resulting from longitudinal–torsional mode coupling. To provide a deeper mechanical interpretation of these avoided crossings, an analytical model with coupled translational and rotational degrees of freedom is developed, offering an analytic perspective on the band-gap formation. The computational predictions of the transmission spectra are benchmarked against experimental data from SLA-printed lattice specimens. We show that capturing the precise transmission characteristics requires a material model that incorporates viscoelastic effects within finite-size simulations. Overall, the results indicate that modest geometric perturbations of classical architected lattices can achieve significant attenuation (up to 20 dB with minimal periodic repetitions), providing a tractable strategy for the design of lightweight, vibration-mitigating structures.