For graphene in one valley, the low energy Hamiltonian writes as:

$$ H_K(q)=\sigma_xq_x+\sigma_y q_y $$

The Hamiltonian on the other valley I remember has two ways of writing(not 100 percent sure), one is: $$ H_{K'}(q)=-\sigma_xq_x+\sigma_y q_y $$

the other is: $$ H_{K'}(q)=-\sigma_xq_x-\sigma_y q_y $$

The two $H_{K'}(q)$ are related to $H_K(q)$ by time reversal operation, I found that the operator for first one is $T_1=\sigma_zK$, for the second is $T_2=-i\sigma_yK$, where $K$ is the complex conjugate operation.

However $T_1^2=I$ while $T_2^2=-I$, which seems to have contradiction, since that spinless fermion has time reversal operator $T^2=I$ and 1/2 spin fermion has $T^2=-I$.

So what is going wrong? What is the time reversal operator for graphene?