We present VertiCoder, a self-supervised representation learning approach for robot mobility on vertically challenging terrain. Using the same pre-training process, VertiCoder can handle four different downstream tasks, including forward kinodynamics learning, inverse kinodynamics learning, behavior cloning, and patch reconstruction with a single representation. VertiCoder uses a TransformerEncoder to learn the local context of its surroundings by random masking and next patch reconstruction. We show that VertiCoder achieves better performance across all four different tasks compared to specialized End-to-End models with 77% fewer parameters. We also show VertiCoder's comparable performance against state-of-the-art kinodynamic modeling and planning approaches in real-world robot deployment. These results underscore the efficacy of VertiCoder in mitigating overfitting and fostering more robust generalization across diverse environmental contexts and downstream vehicle kinodynamic tasks.