The properties of cosmic strings have been investigated in detail for
their implications in early-universe cosmology. Although many variatio
ns of the basic structure have been discovered, with implications for
both the microscopic and macroscopic properties of cosmic strings, the
cylindrical symmetry of the short-distance structure of the string is
generally unaffected. In this paper we describe some mechanisms leadi
ng to an asymmetric structure of the string core, giving the defects a
quasi-two-dimensional character. We also begin to investigate the con
sequences of this internal structure for the microscopic and macroscop
ic physics.