In nature, α-quartz crystals frequently form contact twins – two adjacent crystals with the same chemical structure but different crystallographic orientation, sharing a common lattice plane. As α-quartz crystallises in a chiral space group, such twinning can occur between enantiomorphs with the same handedness or with opposite handedness. Here, we use first principles methods to investigate the effect of twinning and chirality on the bulk and surface phonon spectra, as well as on the topological properties of phonons in α-quartz. We demonstrate that, even though the dispersion appears identical for all twins along all high-symmetry lines and at all high-symmetry points in the Brillouin zone, the dispersions can be distinct at generic momenta for some twin structures. Furthermore, when the twinning occurs between different enantiomorphs, the charges of all Weyl nodal points flip, which leads to mirror symmetric isofrequency contours of the surface arcs on certain surfaces. We show that this allows negative refraction to occur at interfaces between certain twins of α-quartz.