A new family of fully symmetric interpolatory integration rules have been derived. The rules can be used to numerically estimate multidimensional integrals over the surface of a hyper-sphere. The higher degree rules are new, and explicit formulas are given for the weights. The rules are only moderately unstable as the degree of polynomial precision increases. If the rule cost is measured in terms of the number of integrand values, the new rules are efficient, because of their use of fully symmetric sum generators with many components equal to zero. Two randomized error estimation methods have also derived for the rules.