Long-term autonomy requires robust navigation in environments subject to dynamic and static changes, as well as adverse weather conditions. Teach-and-Repeat (T&R) navigation offers a reliable and cost-effective solution by avoiding the need for consistent global mapping; however, existing T&R systems lack a systematic solution to tackle various environmental variations such as weather degradation, ephemeral dynamics, and structural changes. This work proposes LTR$^{2}$, the first cross-modal, cross-platform LiDAR-Teach-and-Radar-Repeat system that systematically addresses these challenges. LTR$^{2}$leverages LiDAR during the teaching phase to capture precise structural information under normal conditions and utilizes 4D millimeter-wave radar during the repeating phase for robust operation under environmental degradations. To align sparse and noisy forward-looking 4D radar with dense and accurate omnidirectional 3D LiDAR data, we introduce a Cross-Modal Registration (CMR) network that jointly exploits Doppler-based motion priors and the physical laws governing LiDAR intensity and radar power density. Furthermore, we propose an adaptive fine-tuning strategy that incrementally updates the CMR network based on localization errors, enabling long-term adaptability to static environmental changes without ground-truth labels. We demonstrate that the proposed CMR network achieves state-of-the-art cross-modal registration performance on the open-access dataset. Then we validate LTR$^{2}$across three robot platforms over a large-scale, long-term deployment (40+ km over 6 months), including challenging conditions such as nighttime smoke. Experimental results and ablation studies demonstrate centimeter-level accuracy and strong robustness against diverse environmental disturbances, significantly outperforming existing approaches.