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We present a detailed study of the internal structure and kinematics of the core of the Sagittarius dwarf spheroidal galaxy (Sgr). Using machine-learning techniques, we have combined the information provided by 3300 RR Lyrae stars, more than 2000 spectroscopically observed stars, and the Gaia second data release to derive the full phase space, i.e. 3D positions and kinematics, of more than $1.2\times10^5$ member stars in the core of the galaxy. Our results show that Sgr has a bar structure $\sim 2.5$ kpc long, and that tidal tails emerge from its tips to form what it is known as the Sgr stream. The main body of the galaxy, strongly sheared by tidal forces, is a triaxial (almost prolate) ellipsoid with its longest principal axis of inertia inclined $43^\circ\pm6^\circ$ with respect to the plane of the sky and axis ratios of 1:0.67:0.60. Its external regions are expanding mainly along its longest principal axis, yet the galaxy conserves an inner core of about $500\times330\times300$ pc that shows no net expansion and is rotating at $v_{\rm rot} = 4.13 \pm 0.16$ ${\rm{ km \ s^{-1}}}$. The internal angular momentum of Sgr forms an angle $θ= 18^\circ\pm6^\circ$ with respect to its orbital angular momentum, meaning that the galaxy is in an inclined prograde orbit around the Milky Way. We compared our results with predictions from $N$-body models with spherical, pressure-supported progenitors and a model whose progenitor is a flattened rotating disk. Only the rotating model, based on preexisting simulations aimed at reproducing the line-of-sight velocity gradients observed in Sgr, was able to reproduce the observed properties in the core of the galaxy.