In this paper, we address the problem of structure-aware shape deformation: We specifically consider deformations that preserve symmetries of the shape being edited. While this is an elegant approach for obtaining plausible shape variations from minimal assumptions, a straightforward optimization is numerically expensive and poorly conditioned. Our paper introduces an explicit construction of bases of linear spaces of shape deformations that exactly preserve symmetries for any user-defined level of detail. This permits the construction of low-dimensional spaces of low-frequency deformations that preserve the symmetries. We obtain substantial speed-ups over alternative approaches for symmetry-preserving shape editing due to i the sub-space approach, which permits low-res editing, ii the removal of redundant, symmetric information, and iii the simplification of the numerical formulation due to hard-coded symmetry preservation. We demonstrate the utility in practice by applying our framework to symmetry-preserving co-rotated iterative Laplace surface editing of models with complex symmetry structure, including partial and nested symmetry.