# Orientifold plane and twisted sector of bosonic string

Consider a bosonic string theory where we truncate the Hilbert space to states that are invariant under an orientifold action $\Omega$, which acts as a worldsheet parity + a space-time reflection, $$\Omega : X[\bar{z},z] \rightarrow -X[z,\bar{z}].$$

Let's compare this to the procedure of orbifolding. In this case, we truncate the Hilbert space to states that are invariant under

$$\Omega : X[\bar{z},z] \rightarrow -X[\bar{z},z],$$

At least in the orbifold case, one can also define a "twisted sector", by demanding

$$X[\sigma + 2\pi]=-X[\sigma]$$

finding the mode expansion, etc. My question is: why can't we define a twisted sector for the orientifold case as well? That would be demanding

$$X[\sigma + 2\pi]=-X[-\sigma]$$

and repeating the same construction of the orbifold case.

In Polchinski's string theory textbook vol 1, it is then claimed that the absence of twisted states for the orientifold imply that the orientifold plane is not dynamical [unlike the D-brane plane case], because then there are no modes tied to the orientifold plane [in a sense, the twisted sector "lives" in the worldvolume of the orbifold fixed plane, in which a twisted sector may be defined].

However he explicitly writes an action for the orientifold plane which is identical to the action of the D-brane with gauge fields equal to zero [eq 8.8.5, pg 280]:

$$S \approx \int d^{p+1} \xi e^{-\phi} \sqrt{-\det G_{ab}}$$

This involves the pullback of the metric to the worldvolume of the brane. Since this is identical to the D-brane case, I don't understand why he calls the orientifold plane "not dynamical" and the D-brane "dynamical".