Representations of SO(4) from two SU2 spins ($(j_L,j_R),j_L+j_R\in \mathbb{Z}$)

Tensor products $V_{j_L}\otimes V_{j_R}$ give the irreducible representations of the Spin(4) double cover. The simultaneous central sign acts by $(-1{)}^{2j_L+2j_R}$, so exactly the integer-sum pairs descend to SO(4). Their dimensions are $(2j_L+1)(2j_R+1)$. Under the diagonal spatial rotation subgroup, the Clebsch-Gordan decomposition for SU2 gives spins from $|j_L-j_R|$ through $j_L+j_R$ in steps of one.