Cook the fusilli pasta according to the pack instructions, or until al dente. Drain and refresh under cold water, and drain again.
In a small bowl, whisk together the olive oil, red wine vinegar, lemon juice and dried oregano, and season to taste.
Halve the cucumber lengthways and, using a teaspoon, scoop out the seeds, then cut into thick slices. Add to the pasta along with the crumbled feta cheese, red onion, tomatoes and fresh basil. Pour over the dressing and gently toss.