<p>In this article we introduce a way to detect symmetry in Mixed-Integer Conic Programming. We present our framework as an extension to the Mixed-Integer Linear Programming case that results in a relatively small formal overhead. To do so, we introduce the concept of symmetry labelings for cones and study their properties. In particular, we establish a relation between such labelings and the orbits of permutations that, when acting on a cone in a properly defined way, leave the latter invariant. This allows us to prove the power of the concept of symmetry labelings for some cones, but at the same time its limitations for others. We also report on computational experiments enabling and disabling symmetry detection in the optimization software package MOSEKon a series of Mixed-Integer Conic Programming problems.</p>