In a jar with a tight fitting lid, combine the olive oil, wine vinegar, pimento, parsley, green onion, garlic, sugar, basil, salt and pepper. Shake vigorously to blend. Set aside.
Cut the block of Cheddar cheese in half lengthwise. Cut crosswise into 1/4 inch slices to form squares. Repeat with cream cheese. Arrange cheese squares alternately in a shallow baking dish with slices standing on edge. You can arrange in a single long row, or 2 parallel rows depending on the size of your dish. Pour the marinade from the jar over the cheese slices, cover and refrigerate for at least 6 hours, or overnight.
To serve, remove slices to a serving platter so they are still arranged in an alternating pattern. Spoon some of the extra marinade over the cheese, and serve with crackers.