Due to its proximity, SN1987A offers a unique opportunity to directly observe the geometry of a stellar explosion as it unfolds. Here we present spectral and imaging observations of SN1987A obtained ∼10,000 days after the explosion with HST/STIS and VLT/SINFONI at optical and near-infrared wavelengths. These observations allow us to produce the most detailed 3D map of Hα to date, the first 3D maps for 44 Ti. The time evolution of Hα shows that it is predominantly powered by X-rays from the ring, in agreement with previous findings. All lines that have sufficient signal show a similar large-scale 3D structure, with a north-south asymmetry that resembles a broken dipole. This structure correlates with early observations of asymmetries, showing that there is a global asymmetry that extends from the inner core to the outer envelope. On smaller scales, the two brightest lines, Hα and [Si I]+[Fe II] 1.644 μm, show substructures at the level of ∼200-1000 km s -1 and clear differences in their 3D geometries. We discuss these results in the context of explosion models and the properties of dust in the ejecta.