  A 2-sphere embedded in the 4-sphere invariant under a circle action is called
a branched twist spin. A branched twist spin is constructed from a 1-knot in
the 3-sphere and a pair of coprime integers uniquely. In this paper, we study,
for each pair of coprime integers, if two different 1-knots yield the same
branched twist spin, and prove that such a pair of 1-knots does not exist in
most cases. Fundamental groups of 3-orbifolds of cyclic type are obtained as
quotient groups of the fundamental groups of the complements of branched twist
spins. We use these groups for distinguishing branched twist spins.
