Polynomials on ${ℝ}^n$ with values in an irreducible ${Spin}_n$-module form a natural representation space for the group ${Spin}_n$. These representations are completely reducible. In the paper, we give a complete description of their decompositions into irreducible components for polynomials with values in a certain range of irreducible modules. The results are used to describe the structure of kernels of conformally invariant elliptic first order systems acting on maps on ${ℝ}^n$ with values in these modules.