As I have said before, add clay to clay, old broken bricks are the best without making any real change to the PH etc.

Dig it thoroughly, incorporate humus rich materials as you go, plus sharp grit

Don't walk on it when it's wet, use boards or (in veg garden) make paths and stick to them...or it will stick to you!!

All of the above. Hi Bob.

Hi John, glad to see you couldn't resist it. Welcome back.

Depending on whether you like hard, back breaking work or not, and on the size of your garden, the alternative is to create deep beds over the surface of the clay, in the deep mulch and no digging technique.
If you pile enough 'muck' and compost on top, you will create a layer of new soil into which you can plant. Despite the initial compaction, most things will root down into the clay.
After a few years, the population of earth worms that will have built up will break up the compaction, and provided you don't walk on the soil when it's wet, and keep putting fresh compost on the surface, you will have few problems.
If you google the 'Ruth Stout' method, you will find lots of information on this technique. (which requires more patience and lots of compost material, but no back breaking digging.)

Shifting lots of compost is also back-breaking work!!