as $$\mathbf{T}^{pn}_{p'n'} = \frac{ik^3}{6\pi} \bar{\bar{\mathrm{O}}}_{n'}^{-1} \alpha^{pn}_{p'n'} \bar{\bar{\mathrm{O}}}_{n}
        \label{eqn:a2T}$$ Eqns. \[eqn:T2a\] and \[eqn:a2T\] are the central results of this study, which we use to express exact higher-order $\boldsymbol{\alpha}$-tensors in units of volume (see Appendix \[appendix:units\] for more details).

$\boldsymbol{\alpha}$-tensor or $\mathbf{T}$-matrix contains information on particle symmetries and conservation laws, as well as complete information on scattering by a particle. Symmetry refers to the invariance upon certain operations, and includes rotational symmetry, mirror symmetry, parity and time-reversal symmetry, and reciprocity. $N$-fold rotational symmetry and mirror symmetry can be confirmed by checking if $\mathbf{T}$-matrix or $\boldsymbol{\alpha}$-tensor is invariant under the transformation. For Onsager reciprocal system, $\mathbf{T}$-matrix elements satisfy the following relationship: $(T^{pn}_{p'n'})^{m}_{m'}=(-1)^{m+m'}(T^{p'n'}_{pn})^{-m'}_{-m}$. This expression has been used to check the accuracy of numerically calculated $\mathbf{T}$-matrix in literature [@Mishchenko1996]. $\boldsymbol{\alpha}$-tensor of reciprocal dipolar particles satisfy: $\alpha_{e1}^{e1} = (\alpha_{e1}^{e1})^\top$, $\alpha_{m1}^{m1} = (\alpha_{m1}^{m1})^\top$, and $\alpha_{e1}^{m1} = (\alpha_{m1}^{e1})^\top$ [@Sersic2011]. The parity operation is given as $\mathbf{r}\rightarrow-\mathbf{r}$. Upon the parity operation, $\mathbf{E}\rightarrow-\mathbf{E}$, $\mathbf{H}\rightarrow\mathbf{H}$, $\nabla\rightarrow-\nabla$. Also, electric multipoles ($a^E_{nm}$ and $b^E_{nm}$) have parity of $(-1)^n$, and magnetic multipoles ($a^M_{nm}$ and $b^M_{nm}$) have $(-1)^{n-1}$. From which, we see that $\mathbf{T}^{En}_{En'}$ and $\mathbf{T}^{Mn}_{Mn'}$ have parity of $(-1)^{n+n'}$, and $\mathbf{T}^{En}_{Mn'}$ and $\mathbf{T}^{Mn}_{En'}$ has $(-1)^{n+n'+1}$. The corresponding $\boldsymbol{\alpha}$-tensor components have the same parity. 