  The union of singular orbits of an effective locally smooth circle action on
the 4-sphere consists of two 2-knots, $K$ and $K^{\prime}$, intersecting at two
points transversely. Each of $K$ and $K^{\prime}$ is called a branched twist
spin. A twist spun knot is an example of a branched twist spin. The Gluck
twists along branched twist spins are studied by Fintushel, Gordon and Pao. In
this paper, we determine the 2-knot obtained from $K$ by the Gluck twist along
$K^{\prime}$. As an application, we give infinitely many pairs of inequivalent
branched twist spins whose complements are homeomorphic.
