A set of tectonophysical methods has been applied to reconstruct the neotectonic stresses of the southeastern Altai Mountains within the Chuya–Kurai depression and its framing structures. It is suggested that at the neotectonic stage, the tectonic structures of the Altai Mountains underwent a transformation of geodynamic conditions—a situation of sublatitudinal horizontal compression with predominant reverse-fault and transpressional movements along faults that existed up to the Neogene was replaced by horizontal shear conditions with a NNE and NE subhorizontal compression axis and WNW and NW extension axis. With such a stress field, the dominant NW-trending faults in this territory are mainly characterized by dextral strike-slip displacements, and NE-trending faults, by sinistral. Submeridionally trending faults that formed at the neotectonic stage show clear signs of extensional structures. A feature of the faults of the Chuya–Kurai depression according to these studies is a fairly small number of megafractures (indicators of shear displacements) in shear zones. This indicates dominant reverse-fault and transpressional movements over large disjunctive structures in the region in the Paleozoic and Mesozoic versus neotectonic shear movements. The results of the research are of practical importance in studying the regional seismicity.