The full crystal structure of the phyllosilicate mineral tuperssuatsiaite, including the positions of the hydrogen atoms in its unit cell, is determined for the first time by using first-principles solid-state methods. From the optimized structure, its infrared spectrum and elastic properties are determined. The computed infrared spectrum is in excellent agreement with the experimental spectrum recorded from a natural sample from Ilímaussaq alkaline complex (Greenland, Denmark). The elastic behavior of tuperssuatsiaite is found to be extremely anomalous and significant negative compressibilities are found. Tuperssuatsiaite exhibits the important negative linear compressibility phenomenon under small anisotropic pressures applied in a wide range of orientations of the applied strain and the very infrequent negative area compressibility phenomenon under external isotropic pressures in the range from 1.9 to 2.4 GPa. The anisotropic negative linear compressibility effect in tuperssuatsiaite is related to the increase of the unit cell along the direction perpendicular to the layers charactering its crystal structure. The isotropic negative area compressibility effect, however, is related to the increase of the unit cell dimensions along the directions parallel to the layers.