Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy

1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.

2) $ a\bar{b}+c\bar{d}=0 $

There is a (component-wise) $S^1$ action on $C$ and let $S$ be the quotient ($S$ is a 3-manifold).

Is $S$ orientable or not ?

Thanks.