Quote by silverwhale
alxm, do you mean by same observables the same expectation values of the observables?

Expectation values, then.
If I understood you right the phase IS the eigenvalue of P. And we obtain this equating the first equation derived from the expectation values to the eigenvalue equation of P.

Switching twice must result in the same function, so the phase factor squared must equal unity, hence 1 or 1.