We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function (Formula presented.) on (Formula presented.) and of its perturbation (Formula presented.), where (Formula presented.) is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere (Formula presented.), thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.