  Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous
in biology. They are considered in textbooks as simple filaments with larger
stiffness. Recent observations of flagellar counterbend, however, show that
induction of curvature in one section of a passive flagellum instigates a
compensatory counter-curvature elsewhere, exposing the intricate role of the
diminutive cross-linking proteins at large-scales. We show that this effect, a
material property of the cross-linking mechanics, modifies the bundle dynamics
non-trivially, and induces a bimodal $L^2-L^3$ length-dependent material
response that departs from the Euler-Bernoulli theory. Hence, the use of
simpler theories to analyse experiments can result in paradoxical
interpretations. Remarkably, the counterbend dynamics instigates counter-waves
in opposition to driven oscillations in distant parts of the bundle, with
potential impact on the regulation of flagellar bending waves. These results
have a range of physical and biological applications, including the empirical
disentanglement of material quantities via counterbend dynamics.
