Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally non-uniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact to the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz- and linearized Euler-equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function, which is solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal mode order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an “inclination rule” that explains the behavior of degenerate eigenvalues at first order is proven.