= Solution
Choose the phase of each inward-pointing ligand sigma donor orbital positive toward its own Mo centre. The eight donors then transform by site permutation. In the usual <D4h molecular point group> class order their characters are
$$
\begin{array}{c|rrrrrrrrrr}
&E&2C_4&C_2&2C_2^\prime&2C_2^{\prime\prime}&i&2S_4&\sigma_h&2\sigma_v&2\sigma_d\\\hline
\chi_{8\sigma}&8&0&0&0&0&0&0&0&4&0
\end{array}
$$
Only $E$ and a chloride-containing vertical mirror plane leave ligand functions individually fixed. For example $\sigma_{xz}$ fixes the four donors at $(\pm a,0,\pm c)$ and exchanges the other four; the four fixed functions retain their signs.
<Molecular character reduction> gives
$$
\boxed{\Gamma_{8\sigma}=A_{1g}\oplus B_{1g}\oplus E_g
\oplus A_{2u}\oplus B_{2u}\oplus E_u.}
$$
The dimensions sum to $1+1+2+1+1+2=8$. Equivalently, take each square's four combinations: the all-in-phase pattern, the alternating x-versus-y pattern, and a two-dimensional pair. Combine corresponding patterns between upper and lower squares to obtain the six irreps above.
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