Mitigating positional bias of language models (LMs) for listwise inputs is a well-known and important problem (e.g., lost-in-the-middle). While zero-shot order-invariant LMs have been proposed to solve this issue, their success on practical listwise problems has been limited. In this work, as a first contribution, we identify and overcome two limitations to make zero-shot invariant LMs more practical: (1) training and inference distribution mismatch arising from modifying positional ID assignments to enforce invariance, and (2) failure to adapt to mixture of order-invariant and sensitive inputs in practical listwise problems. Then, to overcome these issues we propose (1) RoToR, a zero-shot invariant LM for genuinely order-invariant inputs with minimal modifications of positional IDs, and (2) Selective Routing, an adaptive framework that handles both order-invariant and order-sensitive inputs in listwise tasks. On the Lost in the middle (LitM), Knowledge Graph QA (KGQA), and MMLU benchmarks, we show that RoToR with Selective Routing can effectively handle practical listwise input tasks in a zero-shot manner (https://github.com/soyoung97/RoToR)