Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov–Reshetikhin crystals of type \(D^{(1)}_n\) in full generality. We prove the invariance of rigged configurations under the action of the combinatorial R-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type \(D_n^{(1)}\). In addition, we establish that the bijection is a classical crystal isomorphism.