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Lyman-alpha line profiles are a powerful probe of ISM structure, outflowspeed, and Lyman continuum escape fraction. In this paper, we present theLy$\alpha$ line profiles of the COS Legacy Archive Spectroscopic SurveY, asample rich in spectroscopic analogs of reionization-era galaxies. A largefraction of the spectra show a complex profile, consisting of a double-peakedLy$\alpha$ emission profile in the bottom of a damped, Ly$\alpha$ absorptiontrough. Such profiles reveal an inhomogeneous interstellar medium (ISM). Wesuccessfully fit the damped Ly$\alpha$ absorption (DLA) and the Ly$\alpha$emission profiles separately, but with complementary covering factors, asurprising result because this approach requires no Ly$\alpha$ exchange betweenhigh-$N_\mathrm{HI}$ and low-$N_\mathrm{HI}$ paths. The combined distributionof column densities is qualitatively similar to the bimodal distributionsobserved in numerical simulations. We find an inverse relation betweenLy$\alpha$ peak separation and the [O III]/[O II] flux ratio, confirming thatthe covering fraction of Lyman-continuum-thin sightlines increases as theLy$\alpha$ peak separation decreases. We combine measurements of Ly$\alpha$peak separation and Ly$\alpha$ red peak asymmetry in a diagnostic diagram whichidentifies six Lyman continuum leakers in the CLASSY sample. We find a strongcorrelation between the Ly$\alpha$ trough velocity and the outflow velocitymeasured from interstellar absorption lines. We argue that greater vignettingof the blueshifted Ly$\alpha$ peak, relative to the redshifted peak, is thesource of the well-known discrepancy between shell-model parameters anddirectly measured outflow properties. The CLASSY sample illustrates howscattering of Ly$\alpha$ photons outside the spectroscopic aperture reshapesLy$\alpha$ profiles as the distances to these compact starbursts span a largerange.

CLASSY VII Ly\alpha\ Profiles: The Structure and Kinematics of Neutral Gas and Implications for LyC Escape in Reionization-Era Analogs