Hubble Space Telescope imaging and slitless spectroscopy are used to examine where the strong Ly alpha emission escapes from the interstellar medium in the starburst galaxy Mrk 357. An H alpha image shows that the ionized gas is mostly in a global wind, rather than associated with the individual star-forming regions seen in the optical and UV continuum. The Ly alpha emission comes predominantly from the northwest side of the wind structure spatially, and shows a significant redshift relative to the optical lines. Both of these properties are signatures of seeing the line photons backscattered from the far side of a prolate or bipolar starburst wind, fitting both with escape calculations and evidence for winds in high-redshift galaxies with net Ly alpha emission. Scattering is most important within this wind itself, rather than involving a surrounding neutral medium, as shown by the decreasing relative redshift of the line peak from 250 to approximate to 30 km s(-1) between the center and edge of the detected emission. The Ly alpha emission exhibits strong asymmetry in comparison with both the starlight and H alpha structures. These results add to the evidence that kinematics, rather than gas metallicity or dust content, are the dominant effect in determining which galaxies have strong Ly alpha emission, and that powerful ( and perhaps episodic) starbursts are common among Lyman break galaxies as well as those discovered from Ly alpha line emission.