= Solution
The <Bell basis> consists of
$$
|\Phi^\pm\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt2},\qquad
|\Psi^\pm\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt2}.
$$
Let $Z=\sigma_z$, $X=\sigma_x$. The <Pauli operators> $Z\otimes Z$ and $X\otimes X$ commute: each of the two local anticommutations contributes a minus sign, so they cancel. Their eigenvalue table is
$$
\begin{array}{c|cc|cc}
\text{state}&Z\otimes Z&X\otimes X&\text{parity bit}&\text{phase bit}\\\hline
\Phi^+&+1&+1&0&0\\
\Phi^-&+1&-1&0&1\\
\Psi^+&-1&+1&1&0\\
\Psi^-&-1&-1&1&1
\end{array}
$$
To make the bit values literally zero or one rather than signed eigenvalues, the commuting <Bell parity and phase observables> are
$$
\boxed{B_{\rm parity}=\frac{I-Z\otimes Z}{2},\qquad
B_{\rm phase}=\frac{I-X\otimes X}{2}.}
$$
They distinguish all four states jointly. The parity bit records whether the computational bits agree; the phase bit records the relative sign.
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