High-precision proper motions of the globular cluster 47 Tuc have allowed us to measure for the first time the cluster rotation in the plane of the sky and the velocity anisotropy profile from the cluster core out to about 13'. These profiles are coupled with prior measurements along the line of sight (LOS) and the surface brightness profile and fit all together with self-consistent models specifically constructed to describe quasi-relaxed stellar systems with realistic differential rotation, axisymmetry, and pressure anisotropy. The best-fit model provides an inclination angle i between the rotation axis and the LOS direction of 30° and is able to simultaneously reproduce the full three-dimensional kinematics and structure of the cluster, while preserving a good agreement with the projected morphology. Literature models based solely on LOS measurements imply a significantly different inclination angle (i = 45°), demonstrating that proper motions play a key role in constraining the intrinsic structure of 47 Tuc. Our best-fit global dynamical model implies an internal rotation higher than previous studies have shown and suggests a peak of the intrinsic V/σ ratio of ~0.9 at around two half-light radii, with a nonmonotonic intrinsic ellipticity profile reaching values up to 0.45. Our study unveils a new degree of dynamical complexity in 47 Tuc, which may be leveraged to provide new insights into the formation and evolution of globular clusters.