In robotic tasks, changes of reference frames typically do not affect the underlying physical meaning. These are isometric transformations, including translations, rotations, and reflections, called Euclidean group. In this work, we study reinforcement learning and planning tasks that have Euclidean group symmetry. We provide a theory that extends prior work (on symmetry in reinforcement learning, planning, and optimal control) to compact Lie groups and covers them as special cases, and show examples to explain the benefits of equivariance to Euclidean symmetry. We extend the 2D path planning with value-based planning to continuous MDPs and propose a pipeline for equivariant sampling-based planning algorithm with empirical evidence.