  The textures associated with a point defect centered in a circular domain of
a thin film with XY-like ordering have been analyzed. The family of equilibrium
textures, both stable and metastable, can be classified by a new radial
topological number in addition to the winding number of the defect. Chiral
textures are supported in an achiral system as a result of spontaneously broken
chiral symmetry. Among these chiral textures, our theoretical analysis
accurately describes two categories of recently discovered ``reversing spiral''
textures, ones that are energetically stable and metastable.
