A vertically orientated dark matter halo marks a flip of the Galactic disc

Unveiling the 3D shape of the Milky Way's dark-matter halo is critical to understanding its formation history. We created an innovative dynamical model that makes minimal assumptions about the internal dynamical structures and accommodates a highly flexible triaxial DM halo. By applying the method to 6D phase-space data of K-giant stars from LAMOST + Gaia, we robustly determined the 3D dark-matter distribution of the Milky Way out to approximately 50 kpc. We discovered a triaxial, nearly oblate dark-matter halo with q_ ̊m DM = Z/X= 0.92 = Y/X= 0.8 > p_ p_ ̊m DM on average within 50 kpc, where the Z axis is defined perpendicular to the stellar disc. The axes ratio q_ ̊m DM ̊m DM is strongly preferred; the long-intermediate axis plane of the dark-matter halo is unexpectedly vertical to the Galactic disc, yet aligned with the `plane of satellites'. This striking configuration suggests that the Galactic disc (and the inner halo) has flipped, likely torqued by minor mergers, from an original alignment with the outer dark-matter halo and satellite plane, as is supported by Milky Way analogues from Auriga and TNG50. By allowing q_ ̊m DM (r) and p_ ̊m DM (r) to vary with radii, we find tentative evidence that the dark-matter halo is twisted. This agrees alignment with the disc in the inner regions and transitions to a vertical orientation at r≳ 20 kpc, supporting the disc flip scenario prediction. Such disc reorientation is non-trivial, yet its physical mechanism is straightforward to comprehend and naturally originates a vertical satellite plane. Our findings offer a unified framework that links dark-matter halo orientation, satellite alignment, and disc evolution, reinforcing the internal consistency of the Milky Way in the Λ cold dark matter model.