Motivated by the work of McCullough and Trent, we investigate the
z
z
–invariant subspaces of the Hilbert function spaces associated to the Szegő kernels on the open unit disk. In particular, we characterize those kernels for which the the
z
z
–invariant subspaces are hyperinvariant, and (partially) those for which the so-called BLH subspaces are cyclic, obtaining counterexamples to two questions posed by McCullough and Trent.