This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish precise conditions under which these polynomials factor over finite fields, highlighting modular collapses based on prime and composite cycle lengths. We demonstrate that while real-rootedness depends on cycle parity, the combinatorial structure ensures universal log-concavity and unimodality of coefficients. A toggling argument provides a combinatorial proof of unimodality, complementing algebraic methods and offering insights into polynomial stability. These findings bridge combinatorial and algebraic perspectives, contributing to graph-theoretic frameworks with implications in statistical mechanics and information theory.