Plastic characteristics of metals are the result of the motion and interaction of dislocations, line-like crystal defects which move on well-defined crystallographic planes. Full details of evolving dislocation fields can be simulated with the discrete dislocation dynamics (DDD) method. Koopman theory offers a stunning perspective for dislocation-based modelling of crystal plasticity based on DDD data. When approximating the Koopman operator of a physical system it is often beneficial for reduced order modelling and mostly mandatory for interpretability and transferability, that the approximations respect basic physical characteristics of the system. For the spatio-temporal evolution of the slip system–based dislocation fields derived from DDD data, the Koopman operator encompasses spatial interactions and interactions of dislocations on different slip systems. Translational symmetry entails that the former interactions may only depend on relative positions while crystal symmetry entails that the latter interactions only depend on dislocation reaction types between the slip systems. In the current work we combine the group theoretic approach of [1] and the Fourier based approach of [2] to derive an extended dynamic mode decomposition (eDMD) for the dislocation data which yields a block-diagonal Koopman matrix which is simultaneously equivariant with respect to crystal and translational symmetry. We use isotypic projections [1] based on irreducible representations of the full tetrahedral crystal symmetry group T_d to obtain generalized eigenfunctions of the Koopman operator. The matrix valued irreducible representations yield matrix valued isotypic functions and the Koopman matrices on such isotypic subspace become block-matrices. By devising a structure preserving map from the algebra of block matrices to complex matrices we obtain these Koopman block-matrices from data-block matrices by standard methods used in eDMD. We show that the equivariant Koopman matrices yield transferable reduced order models and provide a brief outlook on further challenges for Koopman theory in dislocation modelling.