Generalist robot policies increasingly learn from data collected across diverse tasks and embodiments, yet typically assume a single controller for each robot. This assumption limits transfer because different controllers can assign different physical meanings to the same action and may require different strategies to complete a task. We study how to transfer robot data across both embodiments and controllers without allowing conflicting action interpretations to interfere with learning. Our setting spans six controller types as well as variations in controller gains and compliance. We propose a framework that encodes behavioral interaction under a controller through examples of commanded actions and the resulting robot motions. A transformer context encoder summarizes these interactions, allowing a single policy to adapt in context. Across real-world experiments, our method improves average success by 33 percentage points over naive data pooling, achieves 50\% success on unseen controller modes and 52\% on unseen controller gains, and transfers to tasks with limited controller-specific data. On contact-rich tasks such as USB insertion and surface wiping, behavioral interaction context enables the policy to adapt not only its actions but also its strategy. Our results provide a path toward generalist robot policies that can leverage data collected across diverse action interpretations. For more information, visit \href{https://controller-transfer.github.io}{https://controller-transfer.github.io}.

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