Behold: the mighty spud!

The spud is the part of a cast iron radiator which connects one side of the radiator to the valve (where hot water or steam enters) and the other side to a union joint (where water/steam exits). A deceptively unassuming piece of hardware, the might of the spud lies in its ability to resist efforts to remove it when it has been connected to the radiator for ten thousand years, as many before me have learned. And as I know now!
For every piece of hardware, there is a tool for working with it, or so we are taught. The recommended tool for removal of a spud is called a spud wrench:

The shape of the spud wrench is designed to fit against the two metal ‘buds’ inside of the spud. You then clamp a sturdy pipe wrench to the spud wrench and torque it mightily to loosen and unscrew the spud. Well, that’s what I’ve heard. But what if a spud wrench is as rare as hen’s teeth in your area? I searched high and low at various hardware stores, big and small, for this tool, only to be met with blank stares or the wrong kind of spud wrench altogether (intended for sink and toilet work) or the news that ‘we just sold out.’ Oy.
This is no impediment to the creative and embittered homeowner, though. Time to make shift with the tools on hand. Insert the handle of a smallish pipe wrench (twelve inches, say) into the spud:

Attach a larger pipe wrench to the smaller one:

Now wrench it! But with steady applied force; no jerky violence, please.
This is how I removed the balky spud from one radiator, and I was terribly pleased with myself. That is, I was pleased until I tried to repeat the procedure on the two spuds on a second radiator. This failed, for the buds inside these two spuds were so corroded that they simply snapped off, causing the handle of the twelve-inch wrench to spin impotently in place. Anger…rising…
So what does one do in this spot? One relies on previous unfortunate experience. Earlier, I had goofily turned the nut of one spud in the wrong direction while trying to disconnect the spud from a radiator valve; I cranked the nut away from the radiator and towards the valve. What happened is that I broke the nut, basically; I would up wrenching it off the spud altogether and onto the valve. That was an accident – but I realized that I could perhaps use the same procedure to either remove the nuts from the spuds on this last radiator – in which case I could apply a pipe wrench directly to the outside of the spud – or the spud itself might finally loosen as I cranked the nut and valve away from the radiator. So I took an old, unattached valve, spun the nut onto it, and then held it in place as I deliberately spun the nut in the wrong direction (towards the valve and away from the spud).
And goddamn if that didn’t work. On both spuds. I was able to spin them off (with some elbow grease).

The first success might have been mere happenstance, or shithouse luck, as they say. The second success raised this approach to the level of recommended procedure, and I am happy to share it with you.
For a look at the way normal people remove valves and spuds from cast iron radiators, I recommend this terribly helpful post by Josh at Bungalow Twenty-Three.