The tetragonal heavy-fermion compound CeAuSb$_2$ (space group $P4/nmm$) exhibits incommensurate spin density wave (SDW) order below $T_{N}≈6.5~K$ with the propagation vector $𝐪_A = (δ_A,δ_A,1/2)$. The application of uniaxial stress along the [010] direction induces a sudden change in the resistivity ratio $ρ_a/ρ_b$ at a compressive strain of $ε≈ -0.5$\%. Here we use neutron scattering to show that the uniaxial stress induces a first-order transition to a SDW state with a different propagation vector $(0,δ_B,1/2)$ with $δ_B=0.25$. The magnetic structure of the new (B) phase consists of Ce layers with ordered moments alternating with layers with zero moment stacked along the $c$-axis. The ordered layers have an up-up-down-down configuration along the $b$-axis. This is an unusual situation in which the loss of spatial inversion is driven by the magnetic order. We argue that the change in SDW wavevector leads to Fermi surface reconstruction and a concomitant change in the transport properties.