If using pre-cooked octopus, rinse and drain it well and pat it dry with paper towels. Cut the tentacles into cubes and head, if included, into strips. If using fresh octopus, boil it until tender (see Notes for details), drain it and let it cool until just slightly warm.
Whisk all the dressing ingredients together until well emulsified. Taste and adjust for acidity and seasoning.
Pour the dressing over the octopus and mix well, together with any of the optional ingredients you may be using. Let everything marinate for a good 30 minutes or more.
Serve on a bed of arugula or tender lettuce leaves, with lemon wedges on the side.