In a large bowl, combine the onion, vinegar, oil, lime juice, cane juice, basil, jalapeno, salt and pepper. Stir until the can juice is completely dissolved.
Add the cabbage and stir until it's well coated. Cover and let stand at room temperature for 12 to 24 hours, depending on how strong you want the taste, mixing it once or twice.
Refrigerate until ready to use.
To make a patch of pickled red onions, simply omit the cabbage.
Makes 2 cups