Given a nonzero class $\alpha \in H^{1}(M; \mathbb{Z}/ 2 \mathbb{Z})$ for M a non-orientable closed 3-manifold, can $\alpha$ always be represented by an embedded non-orientable surface $\Sigma \hookrightarrow M$?
More specifically, I have in mind the preimage of a regular value of a map $f$, with $[f] \in [M, \mathbb{R} P^{\infty}]$ the unique homotopy class with $f^{*}(\beta) = \alpha$, $\beta$ a generator of $H^{1}(\mathbb{R}P^{\infty}; \mathbb{Z}/ 2 \mathbb{Z})$. Thinking of $f$ as a map $f: M \rightarrow \mathbb{R}P^{4}$ by cellular approximation and taking the preimage $f^{-1}(\mathbb{R}P^{3})$, we get an embedded surface representing the class. However, I fail to see why this surface need be nonorientable. Is this true in general, or if not is there an easy criterion to check this in examples, perhaps via a computation of the first Stiefel-Whitney class from ambient data?
Edit: If it is not clear as phrased, my question was meant to be interpreted as can the Poincare dual of $\alpha$ be represented by an embedded nonorientable surface.