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Choose amplitude coordinates in which a quarter-turn and a diagonal reflection act as $r(A,B)=(B,-A)$ and $s(A,B)=(-A,B)$. These generate the natural two-dimensional group representation of the dihedral group $D_4$. An equivariant dynamical system satisfies $F(gv)=gF(v)$. The two independent sign changes make $F_A$ odd in $A$ and even in $B$, and $F_B$ odd in $B$ and even in $A$. The quarter-turn equates the two linear coefficients and the two sets of cubic coefficients. No quadratic monomial is allowed. Thus the square-symmetric cubic steady-state normal form isThe bifurcation parameter has been scaled so that the critical linear eigenvalue is $\mu$. Nondegeneracy at cubic order means $a\ne 0$, $a+b\ne 0$ and $a-b\ne 0$. The equilibrium equations factor as $A(\mu -a{A}^2-b{B}^2)=B(\mu -a{B}^2-b{A}^2)=0$. When both components are nonzero, subtracting the two bracketed expressions gives $(a-b)(A^2-B^2)=0$. Therefore the complete leading list of small steady branches is:

Each family is one orbit of the dihedral group. Each nonzero state retains one reflection fixing its amplitude line; its full spatial symmetry depends also on the chosen planforms. Higher-order terms perturb the branch amplitudes but preserve the nondegenerate local classification. For $\mu >0$, an axial branch is asymptotically stable precisely when $a>0$ and $b>a$, while a diagonal branch is asymptotically stable precisely when $a+b>0$ and $b<a$. Every cubic branch lying on $\mu <0$ has a positive radial eigenvalue $-2\mu$, so it is unstable. In the exceptional isotropic cubic case $a=b$, the cubic truncation instead has a circle of equilibria; higher-order anisotropy is then needed, which is why that case was excluded.