I only do supported bike touring, so the "cultural" things are built into the trip. Of course, you are free to partake in as much of that part of the tour as you want to.
The reason why we liked our tour in Andalusia so much is that that Cycling Country, the tour operator we used, is a small business, run by a young couple who hire 1-2 other guides a year. They only do a few trips a year; the one we did took us through the small city where they live. We got to experience their toddler screaming, because she didn't want to go to day care! Not only did we get to see the regular sights in Seville, Granada, and Cordoba, we got a real close up view of the life of regular residents. After the tour, DH and I spent 3 days in Malaga, where we continued walking everywhere and exploring; there were lots of tourists there from the UK, and, as a beach town, it was a little more focused on tourists, but we were still able to get off of the beaten path. We took the bus to the wrong stop and ended up spending the day at the beach, where all of the locals go. So much more interesting than the area set up for tourists.