  Lyman-alpha line profiles are a powerful probe of ISM structure, outflow
speed, and Lyman continuum escape fraction. In this paper, we present the
Ly$\alpha$ line profiles of the 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
Ly$\alpha$ emission profile in the bottom of a damped, Ly$\alpha$ absorption
trough. Such profiles reveal an inhomogeneous interstellar medium (ISM). We
successfully fit the damped Ly$\alpha$ absorption (DLA) and the Ly$\alpha$
emission profiles separately, but with complementary covering factors, a
surprising result because this approach requires no Ly$\alpha$ exchange between
high-$N_\mathrm{HI}$ and low-$N_\mathrm{HI}$ paths. The combined distribution
of column densities is qualitatively similar to the bimodal distributions
observed in numerical simulations. We find an inverse relation between
Ly$\alpha$ peak separation and the [O III]/[O II] flux ratio, confirming that
the covering fraction of Lyman-continuum-thin sightlines increases as the
Ly$\alpha$ peak separation decreases. We combine measurements of Ly$\alpha$
peak separation and Ly$\alpha$ red peak asymmetry in a diagnostic diagram which
identifies six Lyman continuum leakers in the CLASSY sample. We find a strong
correlation between the Ly$\alpha$ trough velocity and the outflow velocity
measured from interstellar absorption lines. We argue that greater vignetting
of the blueshifted Ly$\alpha$ 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 Ly$\alpha$ photons outside the spectroscopic aperture reshapes
Ly$\alpha$ profiles as the distances to these compact starbursts span a large
range.
