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$\newcommand{\Z}{\mathbb{Z}}$ I'm not entirely sure that this is what you're looking for, but it seems like the total symmetry is a $G=\Z_2\rtimes \Z_2^T$, with antiunitary $\Z_2^T$, so the number of SPT phases is inside [1] $$\mathrm{H}^{d+1}(G,U(1)) = \begin{cases} \Z_2^{(d+2)/2} & d = 0 \mod 2 \\ \Z^{(d+1)/2}_{2} & d = 1 \mod 2 \end{cases}$$ So ...