  In nature, $\alpha$-quartz crystals frequently form contact twins - two
adjacent crystals with the same chemical structure but different
crystallographic orientation, sharing a common lattice plane. As
$\alpha$-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-principle 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 $\alpha$-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. We show that this allows negative refraction to occur at interfaces
between certain twins of $\alpha$-quartz.
