We investigate the Lyα and Lyman continuum (LyC) properties of the Sunburst Arc, a z = 2.37 gravitationally lensed galaxy with a multiply imaged, compact region leaking LyC and a triple-peaked Lyα profile indicating direct Lyα escape. Non-LyC-leaking regions show a redshifted Lyα peak, a redshifted and central Lyα peak, or a triple-peaked Lyα profile. We measure the properties of the Lyα profile from different regions of the galaxy using R ∼ 5000 Magellan/Magellan Echellette spectra. We compare the Lyα spectral properties to LyC and narrowband Lyα maps from Hubble Space Telescope imaging to explore the subgalactic Lyα−LyC connection. We find strong correlations (Pearson correlation coefficient r > 0.6) between the LyC escape fraction and Lyα (1) peak separation v_sep, (2) ratio of the minimum flux density between the redshifted and blueshifted Lyα peaks to continuum flux density f_min/f_cont, and (3) equivalent width. We favor a complex H i geometry to explain the Lyα profiles from non-LyC-leaking regions and suggest two H i geometries that could diffuse and/or rescatter the central Lyα peak from the LyC-leaking region into our sight line across transverse distances of several hundred parsecs. Our results emphasize the complexity of Lyα radiative transfer and its sensitivity to the anisotropies of H i gas on subgalactic scales. Large differences in the physical scales on which we observe spatially variable direct-escape Lyα, blueshifted Lyα, and escaping LyC photons in the Sunburst Arc underscore the importance of resolving the physical scales that govern Lyα and LyC escape.