  We investigate constraints on embeddings of a non-orientable surface in a
$4$-manifold with the homology of $M \times I$, where $M$ is a rational
homology $3$-sphere. The constraints take the form of inequalities involving
the genus and normal Euler class of the surface, and either the
Ozsv\'ath--Sazb\'o $d$-invariants or Atiyah--Singer $\rho$-invariants of $M$.
One consequence is that the minimal genus of a smoothly embedded surface in
$L(2p,q) \times I$ is the same as the minimal genus of a surface in $L(2p,q)$.
We also consider embeddings of non-orientable surfaces in closed $4$-manifolds.
