Many or most gardeners agree with Chelle. You don't have to turn it under. I would agree this far: you don't need to turn it under if your soil is already half-decent and you are just maintaining its organic content and fertility.
Also, maybe there is no need if you are patient and don't want to do a lot of pick-and-hoe work.
But I think the compost will break down much faster if covered with soil, and the soil will improve overnight instead of over a year or three, if you turn it and break up the clods and mix it.
My own practice, when breaking up heavy clay that lacks worms and lacks aeration and lacks drainage, is to break up the clay while slightly moist, using a pick or mattock. I go at least 6" deep, usually 12" or more. Than I spread the compost, manure or mulch-like product. (For me, shredded conifer bark.)
Then I turn the compost under the clay I just broke up, to get it down there a s soon as possible and aerate the clay so that something can push roots into the clay and hopefully keep it aerated.
Almost everyone says that you can "just wait" and worms will do the heavy work for you. In my yard, I'm still waiting for the worms to come. The only things that keep the clay from reverting to clay is what I turn under and any roots that follow it.
I've found that they are pretty right, even in moderately heavy clay soil, once the initial improvement has occurred and the soil is light enough that at least weeds can thrive in it.
Maintaining soil capable of supporting weeds and some other plants does seem to work by the "top-down" or "trickle-down" method.
Lot's of people have said that they can get good growth by "lasagna layering" lots of compost on top of what they call "heavy clay" and growing plants in the compost. After a few years (or less?) they say that the underlying "clay" has become great loam.
I haven't experienced that, partly because I've never had a 4" layer of compost to put anywhere. And maybe people call soil "clay" even when it is only 40% or 50% clay. Lots of methods would probably wok on clay soil that is getting to be close to "clay loam soil".
I'm guessing that I have somewhere around 60-80% clay in most of my yard, and 90% in one nasty bed.
