  Whenever the elastic energy of a solid depends on magnetic field, there is a
magnetostrictive response. Field-linear magnetostriction implies piezomagnetism
and vice versa. Here, we show that Mn$_3$Sn, a non-collinear antiferromanget
with Weyl nodes, hosts a large and almost perfectly linear magnetostriction
even at room temperature. The longitudinal and transverse magnetostriction,
with opposite signs and similar amplitude are restricted to the kagome planes
and the out-of-plane response is negligibly small. By studying four different
samples with different Mn:Sn ratios, we find a clear correlation between the
linear magnetostriction, the spontaneous magnetization and the concentration of
Sn vacancies. The recently reported piezomagnetic data fits in our picture. We
show that linear magnetostriction and piezomagnetism are both driven by the
field-induced in-plane twist of spins. A quantitative account of the
experimental data requires the distortion of the spin texture by Sn vacancies.
We find that the field-induced domain nucleation within the hysteresis loop
corresponds to a phase transition. Within the hysteresis loop, a concomitant
mesoscopic modulation of local strain and spin twist angles, leading to
twisto-magnetic stripes, arises as a result of the competition between elastic
and magnetic energies.
