  Stars slingshotted by the supermassive black hole at the Galactic centre will
escape the Milky Way so quickly that their trajectories will be almost straight
lines. Previous works have shown how these `hypervelocity stars' are
subsequently deflected by the gravitational field of the Milky Way and the
Large Magellanic Cloud (LMC), but have neglected to account for the reflex
motion of the Milky Way in response to the fly by of the LMC. A consequence of
this motion is that the hypervelocity stars we see on the outskirts of the
Milky Way today were ejected from where the Milky Way centre was hundreds of
millions of years ago. This change in perspective causes large apparent
deflections in the trajectories of the hypervelocity stars, which are of the
same order as the deflections caused by the gravitational force of the Milky
Way and LMC. We quantify these deflections by simulating the production of
hypervelocity stars in an isolated Milky Way (with a spherical or flattened
dark matter halo), in a fixed-in-place Milky Way with a passing LMC, and in a
Milky Way which responds to the passage of the LMC. The proper motion precision
necessary to measure these deflections will be possible with the combination of
Gaia with the proposed GaiaNIR successor mission, and these measurements will
unlock the hypervelocity stars as probes of the shape of the Milky Way, the
mass of the LMC, and of the dance of these two galaxies.
