  Let $M$ be a compact $4$-manifold and let $S$ and $T$ be embedded $2$-spheres
in $M$, both with trivial normal bundle. We write $M_S$ and $M_T$ for the
$4$-manifolds obtained by the Gluck twist operation on $M$ along $S$ and $T$
respectively. We show that if $S$ and $T$ are concordant, then $M_S$ and $M_T$
are $s$-cobordant, and so if $\pi_1(M)$ is good, then $M_S$ and $M_T$ are
homeomorphic. Similarly, if $S$ and $T$ are homotopic then we show that $M_S$
and $M_T$ are simple homotopy equivalent. Under some further assumptions, we
deduce that $M_S$ and $M_T$ are homeomorphic. We show that additional
assumptions are necessary by giving an example where $S$ and $T$ are homotopic
but $M_S$ and $M_T$ are not homeomorphic. We also give an example where $S$ and
$T$ are homotopic and $M_S$ and $M_T$ are homeomorphic but not diffeomorphic.
