In this paper we report on observations of unusual linear defects forming spontaneously in polar free-standing smectic-C* films near the temperatures of thinning transitions. At high temperature a periodic structure of defects becomes the ground state of the system. We found that the defects are characterized by continuous rotation of the molecular orientation with a change of the sense of the rotation across the defects. We develop a simple theoretical model that describes the observed behavior. The structure of the defects is governed by the competition between two-dimensional quadratic and linear orientational elasticity. The proposed model explains the origin of the linear defects, the periodic structure and their transformation with temperature and chirality of the liquid crystal.