= Solution
The identity <permutation> fixes every <multivariate polynomial>. With permutation composition $(\sigma\tau)(i)=\sigma(\tau(i))$, substitution gives
$$
[\sigma\cdot(\tau\cdot f)](x_1,\ldots,x_n)
=f(x_{\sigma(\tau(1))},\ldots,x_{\sigma(\tau(n))})
=[(\sigma\tau)\cdot f](x_1,\ldots,x_n).
$$
Thus this is a left <variable-permutation action on polynomials>. The substitution acts on variable names; using the displayed definition avoids reversing the composition order.
For the given <multivariate polynomial>, the <group orbit> corresponds to the three partitions of four indices into two unordered pairs:
$$
\boxed{\operatorname{Orb}(f)=\{x_1x_2+x_3x_4,\ x_1x_3+x_2x_4,\ x_1x_4+x_2x_3\}.}
$$
These three <polynomials> are distinct, and every pairing occurs under a <permutation>. The <orbit-stabilizer theorem> gives
$$
\boxed{|(S_4)_f|=24/3=8.}
$$
The eight elements of the <stabiliser subgroup> independently interchange the members of each pair, and may also interchange the two pairs.
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