Due to its proximity, SN. 1987A offers a unique opportunity to directly observe the geometry of a stellar explosion as it unfolds. Here we present spectral and imaging observations of SN. 1987A obtained similar to 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 Ha to date, the first 3D maps for [Ca II] lambda lambda 7292, 7324, [O I] lambda lambda 6300, 6364, and Mg. II lambda lambda 9218, 9244, as well as new maps for [Si I]+[Fe II] 1.644 mu m and He I 2.058 mu m. A comparison with previous observations shows that the [Si I]+[Fe II] flux and morphology have not changed significantly during the past ten years, providing evidence that this line is powered by Ti-44. The time evolution of Ha 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, Ha and [Si I]+[Fe II] 1.644 mu m, show substructures at the level of similar to 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.