It is by now well established that in antiferromagnetic γ-Fe, stabilized in the form of precipitates in a Cu matrix or by epitaxial growth on an appropriate substrate, magnetic and/or crystalline symmetries are broken. Little is known, however, on the physical effects driving the symmetry reduction, and on the interplay of crystalline and magnetic symmetry breaking. We have used a recently developed unconstrained vector-field description of noncollinear magnetism, implemented in an ab initio spin-density-functional code, to search for the magnetic and crystalline structure of γ-Fe, stabilized by different types of constraints. We show that in near face-centered-cubic γ-Fe, stabilized by three-dimensional constraints, the magnetic ground state is a spin-spiral with propagation vector q⃗=2π/a×(0.2,0,1) at an equilibrium atomic volume of Ω=10.63 ų, very close to the propagation vector qexp=2π/a×(0.1,0,1), determined experimentally, but at considerably lower volume than the atomic volume of the γ-Fe precipitates in Cu on which the experiments were performed (Ω=11.44Å ³). At these larger volumes our calculations predict an helical spin solution at q⃗=2π/a×(0,0,0.6) to be the ground state. Epitaxially stabilized γ-Fe is found to be unstable against both tetragonal distortion as well as monoclinic shear deformation, and the structural distortions suppress the formation of spin-spiral states, in agreement with experimental observations on Fe/Cu(100) films.