Filamentous viruses like influenza and torovirus often display systematic bends and arcs of mysterious physical origin. We propose that such viruses undergo an instability from a cylindrically symmetric to a toroidally curved state. This “toro-elastic” state emerges via spontaneous symmetry breaking under prestress due to short range spike protein interactions magnified by surface topography. Once surface stresses are sufficiently large, the filament buckles and the curved state constitutes a soft mode that can potentially propagate through the filament's material frame around a Mexican-hat-type potential. In the mucus of our airways, which constitutes a soft, porous 3D network, glycan chains are omnipresent and influenza's spike proteins are known to efficiently bind and cut them. We next show that such a non-equilibrium enzymatic reaction can induce spontaneous rotation of the curved state, leading to a whole body reshaping propulsion similar to – but different from – eukaryotic flagella and spirochetes.