Place the cabbage in a large bowl and toss it with the salt, using your hands to press down firmly on the cabbage to encourage the cabbage to release some of its moisture and better absorb the salt. Transfer the cabbage to a colander and let it sit for 20 minutes to drain.
Using paper towels or a kitchen towel, squeeze as much liquid as possible from the cabbage and place it in a clean, large bowl. Toss with the mangoes, cilantro, lime juice, vinegar and sugar. Let the coleslaw sit at room temperature for 15 minutes before serving.