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We apply nonperturbative renormalization to bilinears composed of improved staggered fermions. We explain how to generalize the method to staggered fermions in a way that is consistent with the lattice symmetries and introduce a new type of lattice bilinear that transforms covariantly and avoids mixing. We derive the consequences of lattice symmetries for the propagator and vertices. We implement the method numerically for hypercubic-smeared (HYP-smeared) and asqtad valence fermion actions, using lattices with asqtad sea quarks generated by the MILC Collaboration. We compare the nonperturbative results so obtained to those from perturbation theory, using both scale-independent ratios of bilinears (of which we calculate 26) and the scale-dependent bilinears themselves. Overall, we find that one-loop perturbation theory provides a successful description of the results for HYP-smeared fermions if we allow for a truncation error of roughly the size of the square of the one-loop term (for ratios) or of size Q(1) X a² (for the bilinears themselves). Perturbation theory is, however, less successful at describing the nonperturbative asqtad results. [ABSTRACT FROM AUTHOR]