Bring a large pot of generously salted water to a rolling boil. Add the rotini or fusilli and cook according to package instructions until al dente, usually 9 to 11 minutes.
Drain the pasta in a colander and rinse under cold water until completely cool. Shake off excess water and set aside in a large mixing bowl.
Halve the cherry tomatoes and add to the bowl. Dice the cucumber, thinly slice the red onion, halve the kalamata olives, slice the roasted red peppers into strips, and quarter the artichoke hearts.
If the raw red onion is too sharp, soak the slices in cold water for 10 minutes, then drain before adding to the salad. Chop the parsley and set aside to add fresh at the end.
In a mason jar or small bowl, combine the minced garlic, extra virgin olive oil, red wine vinegar, lemon juice, dried oregano, salt, and pepper. Seal and shake vigorously for about 20 seconds until combined and slightly thickened.
Taste and adjust the salt or vinegar if needed; the dressing should taste slightly sharper than the final salad since the pasta will absorb some acidity.
Add the cucumber, red onion, olives, roasted red peppers, and artichoke hearts to the bowl with the pasta and tomatoes. Pour the dressing over the top and toss thoroughly with tongs or a large spoon until evenly coated.
Gently fold in the crumbled feta cheese last so it doesn't break apart into the dressing. Stir in the chopped parsley.
Cover the bowl and refrigerate for at least 30 minutes before serving, though a few hours gives the flavors more time to meld. Give the salad one more toss right before serving, as the dressing tends to settle at the bottom.