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Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space $${{\mathbb R}^m}$$. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on $${{\mathbb R}^m}$$. Fix the direct sum $${{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}$$. In this article, we will study the decomposition of the space $${{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}$$ of spherical monogenics of order n under the action of Spin( p) × Spin( q). As a result, we obtain a Spin( p) × Spin( q)-invariant orthonormal basis for $${{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}$$. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space $${{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}$$. [ABSTRACT FROM AUTHOR]