Rather than responding to Alex's question "What exactly does Marx mean when he inverted Hegel's dialectic?" by analysing in detail how Hegel presents the relation of mind and matter in his works, I would like to quote just a couple of his more "mystical" statements on this relation and then discuss in fairly lay terms what objective idealism means to me.
In talking of the history of philosophy, in §13 of the Shorter Logic, Hegel says:
For these thousands of years the same Architect has directed the work: and that Architect is the one living Mind whose nature is to think, to bring to self-consciousness what it is, and, with its being thus set as object before it, to be at the same time raised above it, and so to reach a higher stage of its own being.
And towards the end of the Shorter Logic (§239):
The truth is that Nature is the creation of Spirit, and it is Spirit itself which gives itself a presupposition in Nature.
In the early 1980s, at the time I was doing my first study of Hegel's Logic, I was a technician in the Physics Department at a British Polytechnic. There was a lecturer who specialised in the physics of the human body. He was also a Jehovah's Witness. I derived great pleasure by listening in on his lectures because it fascinated me how someone who believed that the Soul resided in the human blood and survived death, etc., could teach the physics of Life without deviating from the conventional natural science syllabus. During the same period, I had some discussions with Iranian students who were traditional Muslims and studied science at the Polytechnic. They accepted the quantum character of sub-stomic entities as described in modern physics, but held that an electron's every spin and leap was executed according to the Will of Allah.
Old-fashioned Christianity (or new-fashioned) and Islam are both instances of objective idealism, albeit much cruder than Hegel and contributing nothing to human progress in this century. The priority given to Spirit is indeed "in the last instance", and is compatible with acceptance of science. In the case of Religion, this objective idealism leaves the door open in addition to absolutely non-scientific theories which rest on direct access to knowledge of the Will of God not revealed in practice, perception, introspection or critique of scientifc thought and observation.
In the case of Hegel it goes roughly like this:
This last dot-point Hegel would not agree with. The immediate is richer, he would say, from the standpoint of its content than any given stage of the Idea. But in practice, in consideration of social, historical and natural phenomena, Hegel falls into this trap.
This is somewhat like the idealism I observed among my fellow engineering students in the first years of University. Having learnt the basic laws of inertia, friction, gravity, etc., they would assert things about the behaviour of natural systems which manifestly contradicted day-to-day experience (errors reflecting their incomplete knowledge of mechanics), so confident were they that they knew the "laws" that natural objects "obeyed". Indeed, nature "obeying laws" is an expression which reflects this conception, which, like all heuristic conceptions has its merits (it would be almost impossible to write science without having recourse to such a phrase), but is patently idealist "in the final analysis".
Epistemology reflects and expresses method. Laws are objective, but only ever approximate reality. The reproduction of the immediate in thought-forms can never be exhaustive. Nature is essentially and infinitely richer than human knowledge. It is a bottomless well of knowledge. For the idealist of either kind, beyond the realm of knowledge, if not within it, lies God.
This assumption, that just beyond the bounds of knowledge lies God, Allah, Spirit, Gaia, the next chapter in the textbook or the advice of the Comrade Branch Secretary, is idealism.
For instance, Engels says (in Ludwig Feuerbach, Part IV):
the philosophy of history, particularly as represented by Hegel, recognises that the ostensible and also the really operating motives of people who act in history are by no means the ultimate causes of historical events; that behind these motives are other motive powers, which have to be discovered. But it does not seek these powers in history itself, it imports them rather from outside, from philosophical ideology, into history.
In a sense, the "turning of Hegel upon his feet" is just a small "correction", a matter of detail. The whole of natural science, Marxist political economy and any body of genuine knowledge can be asserted without bothering about this little matter of detail. The problem comes when our knowledge proves inadequate in the face of experience and we need to critique knowledge in order to comprehend objective reality, when Comrade Branch Secretary's explanation doesn't stand up, when science fails to provide an adequate explanation. The problem is also further complicated by the frequent gap between theory and practice which arises in such moments of crisis - even the avowed Marxist may prefer to believe Comrade Branch Secretary and be simply blind to the patent inadequacy of Comrade Branch Secretary's explanation. It is well known in perceptual psychology that we tend to see only what we can recognise or make sense of.