Large language models (LLMs) are increasingly used for recommendation reranking, but their listwise predictions can depend on the order in which candidates are presented. This creates a mismatch between the set-based nature of recommendation and the sequence-based computation of decoder-only LLMs, where permuting an otherwise identical candidate set can change item scores and final rankings. Such order sensitivity makes LLM-based rerankers difficult to rely on, since rankings may reflect prompt serialization rather than user preference. We propose InvariRank, a permutation-invariant listwise reranking framework that addresses this dependence at the architectural level. InvariRank blocks cross-candidate attention with a structured attention mask and negates position-induced scoring changes through shared positional framing under Rotary Positional Embeddings (RoPE). Combined with a listwise learning-to-rank objective, the model scores all candidates in a single forward pass, avoiding permutation-based invariance training objectives that require multiple permutations of a candidate set. Experiments on recommendation benchmarks show that InvariRank maintains competitive ranking effectiveness while producing stable rankings across candidate permutations. The results suggest that architectural invariance is a practical route to reliable and efficient LLM-based recommendation reranking. The source code is at https://github.com/ejbito/InvariRank.