Abstract Lyα line profiles are a powerful probe of interstellar medium (ISM) structure, outflow speed, and Lyman-continuum escape fraction. In this paper, we present the Lyα line profiles of the Cosmic Origins Spectrograph (COS) Legacy Archive Spectroscopic SurveY, a sample rich in spectroscopic analogs of reionization-era galaxies. A large fraction of the spectra show a complex profile, consisting of a double-peaked Lyα emission profile in the bottom of a damped, Lyα absorption trough. Such profiles reveal an inhomogeneous ISM. We successfully fit the damped Lyα absorption and the Lyα emission profiles separately, but with complementary covering factors, a surprising result because this approach requires no Lyα exchange between high-N _{H i} and low-N _{H i} paths. The combined distribution of column densities is qualitatively similar to the bimodal distributions observed in numerical simulations. We find an inverse relation between Lyα peak separation and the [O iii]/[O ii] flux ratio, confirming that the covering fraction of Lyman-continuum-thin sightlines increases as the Lyα peak separation decreases. We combine measurements of Lyα peak separation and Lyα red peak asymmetry in a diagnostic diagram, which identifies six Lyman-continuum leakers in the COS Legacy Archive Spectrocopy SurveY (CLASSY) sample. We find a strong correlation between the Lyα trough velocity and the outflow velocity measured from interstellar absorption lines. We argue that greater vignetting of the blueshifted Lyα peak, relative to the redshifted peak, is the source of the well-known discrepancy between shell-model parameters and directly measured outflow properties. The CLASSY sample illustrates how scattering of Lyα photons outside the spectroscopic aperture reshapes Lyα profiles because the distances to these compact starbursts span a large range.