  This paper delves into the connection between flat and curvilinear
magnetization dynamics. For this, we numerically study the evolution of the
magnon spectrum of rectangular waveguides upon rolling its cross-section up to
a full tube. Magnon spectra are calculated over a wide range of magnetization
states using a finite-element dynamic-matrix method, which allows us to trace
the evolution of the magnon frequencies and several critical magnetic fields
with increasing curvature. By analyzing the parity of the higher-order magnon
modes, we find a curvature-induced mode hetero-symmetry that originates from a
chiral contribution to the exchange interaction and is related to the Berry
phase of magnons in closed loops. Importantly, this curvature-induced parity
loss has profound consequences for the linear coupling between different
propagating magnons, allowing for hybridization between initially orthogonal
modes. In this context, we demonstrate the integral role of edge modes in
forming the magnon spectrum in full tubes. Our findings provide new theoretical
insights into curvilinear magnetization dynamics and are relevant for
interpreting and designing experiments in the field.
