Table 1.

Symmetry broken states and the remaining symmetries for all possible translation-symmetric gapped states at $\mathbf{\nu }=1,2,3$.

$\nu =2$	Example of the state	Symmetry
SP	$∣\uparrow {\mathbf{K}}_{+}⟩\otimes ∣\uparrow {\mathbf{K}}_{-}⟩$	$U{\left(1\right)}_z$, $U{\left(1\right)}_v$, $\mathcal{T}$
VP	$∣\uparrow {\mathbf{K}}_{+}⟩\otimes ∣\downarrow {\mathbf{K}}_{+}⟩$	$SU\left(2\right)$, $\mathcal{T}$
SVL	$∣\uparrow {\mathbf{K}}_{+}⟩\otimes ∣\downarrow {\mathbf{K}}_{-}⟩$	$U{\left(1\right)}_z$, $U{\left(1\right)}_v$, ${\mathcal{T}}^{\prime }$
IVC	$\left(\right.∣\uparrow {\mathbf{K}}_{+}⟩+e^{i\theta }∣\uparrow {\mathbf{K}}_{-}⟩\left)\right.\otimes \left(\right.∣\downarrow {\mathbf{K}}_{+}⟩+e^{i\theta }∣\downarrow {\mathbf{K}}_{-}⟩\left)\right.$	$SU\left(2\right)$, $\mathcal{T}$
SIVCL	$\left(\right.∣\uparrow {\mathbf{K}}_{+}⟩+e^{i\theta }∣\downarrow {\mathbf{K}}_{-}⟩\left)\right.\otimes \left(\right.∣\downarrow {\mathbf{K}}_{+}⟩+e^{i\theta }∣\uparrow {\mathbf{K}}_{-}⟩\left)\right.$	$U{\left(1\right)}_z$, ${\mathbb{Z}}_2^{xv}$, $\mathcal{T}$

$\nu =1,3$	Example of the state	Symmetry
SVP	$∣\uparrow {\mathbf{K}}_{+}⟩$	$U{\left(1\right)}_z$, $U{\left(1\right)}_v$
SPIVC	$∣\uparrow {\mathbf{K}}_{+}⟩+e^{i\theta }∣\uparrow {\mathbf{K}}_{-}⟩$	$U{\left(1\right)}_z$, $\mathcal{T}$
SVLIVC	$∣\uparrow {\mathbf{K}}_{+}⟩+e^{i\theta }∣\downarrow {\mathbf{K}}_{-}⟩$	${\mathbb{Z}}_2^{zv}$, ${\mathcal{T}}^{\prime }$

The similar table with the form of $M\left(\mathbf{k}\right)$ and symmetry generators is in the Supplementary Table 1. Here, $\mathcal{T}$ is the spinless time-reversal $\mathcal{T}={\tau }_x\mathcal{K}$ squaring to $+1$ whereas ${\mathcal{T}}^{\prime }$ is the spinful time-reversal ${\mathcal{T}}^{\prime }=i{\sigma }_y\mathcal{T}$ squaring to $-1$ (with $\mathcal{K}$ denoting complex conjugation). $U{\left(1\right)}_{x,y,z}^{\theta }=e^{i\theta {\sigma }_{x,y,z}∕2}$ denotes spin rotation around the $x,y,z$ axis by an angle $\theta$ whereas $U{\left(1\right)}_v^{\theta }=e^{i\theta {\tau }_z∕2}$ denotes rotation in the valley $x-y$ plane by an angle $\theta$. Finally, ${\mathbb{Z}}_2^{z,v}$ is generated by the combined rotation $U{\left(1\right)}_z^{\pi }U{\left(1\right)}_v^{\pi }$