We consider the nonconvex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank-one terms. Use is made of the rich symmetry structure to construct infinite families of critical points represented by Puiseux series in the problem dimension, and so obtain precise analytic estimates on the objective function value and the Hessian spectrum. The results enable an analytic characterization of various obstructions to local optimization methods, revealing, in particular, a complex array of saddles and minima that differ in their symmetry, structure, and analytic properties. A notable phenomenon, observed for all critical points considered, concerns the index of the Hessian increasing with the objective function value.