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We characterise, in the setting of the Kodaira-Spencer deformation theory,the twistor spaces of (co-)CR quaternionic manifolds. As an application, weprove that, locally, the leaf space of any nowhere zero quaternionic vectorfield on a quaternionic manifold is endowed with a natural co-CR quaternionicstructure. Also, for any positive integers $k$ and $l$, with $kl$ even, weobtain the geometric objects whose twistorial counterparts are complexmanifolds endowed with a conjugation without fixed points and which preservesan embedded Riemann sphere whose normal bundle is $l$ times the line bundle ofChern number $k$. We apply these results to prove the existence of naturalclasses of co-CR quaternionic manifolds.

On the twistor space of a (co-)CR quaternionic manifold