Years ago we lived in upstate South Carolina, true red clay, "Adobe brick" country. That's where I started composting EVERYTHING. I would rip up cereal boxes, bill envelopes, lots of different stuff. I put rinds and harder parts of vegetables and fruit in the blender before adding it to the soil. A regular rototiller bounced instead of tilling, but a friend had a big Troybilt Pony tiller that she lent us. It was able to break up the clay.
We have gray clay and sand here in eastern NC. Instead of breaking my back trying to dig I put down sheets of cardboard, flattened boxes, where I plan to dig. It works even better if I put a layer of leaves under the cardboard. Within a month or two the soil under what's left of the cardboard is friable, and contains lots of worms and millipedes (pill bugs, roll polys).