The \( 3j \) symbols obey the symmetry relation $$ \left(\begin{array}{ccc} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3\end{array}\right)=(-1)^{p}\left(\begin{array}{ccc} j_a & j_b & j_c \\ m_a & m_b & m_c\end{array}\right), $$ with \( (-1)^p=1 \) when the columns \( a,b, c \) are even permutations of the columns \( 1,2,3 \), \( p=j_1+j_2+j_3 \) when the columns \( a,b,c \) are odd permtations of the columns \( 1,2,3 \) and \( p=j_1+j_2+j_3 \) when all the magnetic quantum numbers \( m_i \) change sign. Their orthogonality is given by $$ \sum_{j_3m_3}(2j_3+1)\left(\begin{array}{ccc} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3\end{array}\right)\left(\begin{array}{ccc} j_1 & j_2 & j_3 \\ m_{1'} & m_{2'} & m_3\end{array}\right)=\delta_{m_1m_{1'}}\delta_{m_2m_{2'}}, $$ and $$ \sum_{m_1m_2}\left(\begin{array}{ccc} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3\end{array}\right)\left(\begin{array}{ccc} j_1 & j_2 & j_{3'} \\ m_{1} & m_{2} & m_{3'}\end{array}\right)=\frac{1}{(2j_3+1)}\delta_{j_3j_{3'}}\delta_{m_3m_{3'}}. $$