Squeal in drum brakes is commonly caused by friction linings that are too hard (intended to increase wear life). You used to be able to choose between soft linings for quiet or hard linings for longer life, but I haven't seen this choice advertised for decades.

Turning the drums to get a smoother surface may well eliminate the squeal, or reduce it somewhat. Also a few repeated panic stops in a row from highway speed may rub a glazed surface off of the linings to stop the squeal. Worth a try before having the drums machined. Around here most of the "serious" auto parts stores (NAPA for instance) have a brake lathe to turn drums and rotors, as the turning process is still cheaper than replacement drums and rotors. There is not a lot of material allowed for machining on the MGA drums, perhaps 0.040" (dia) for rear drums and 0.060" (dia) for front drums. Take out more material and the adjusters won't work. As such, and considering the current replacement cost of the drums, I would only have them turned as a last resort.

Some years ago I had a different problem. When I was autocrossing regularly (every week end) I would use the brakes heavily for short bursts from speed. I'm sure this would heat the drums considerably. Eventually this repeated heat cycling will change the grain structure of the metal at the friction surface, similar to flame hardening. Some carbon from the iron will coalesce at the grain boundaries to create carbon nodules (which might be see under a microscope). This makes for a wear resistant surface, but cause other problems.

For instance, I once had a front brake locking up at low speed (about 10-mph) with light pedal application. This was bad news for competition. No matter if I eased off the brakes at the anticipated moment, it would lock up anyway. Cleaning the drums and shoes, sanding the parts, and replacing shoes made no difference. Eventually I gave in and had the drums turned, even though they were perfectly round and smooth, and that cured the problem.

A point of interest was that the cutting tool would chatter like crazy with a shallow cut, so the technician had to cut 0.015" deep to stop the tool from chattering. I recon the hardened surface must have been at least 0.010" deep. That was the only time I have ever had the drums turned in 240,000 miles. They do not otherwise seem to wear out, as long as you keep good linings on the shoes.

On 9/24/2014, Barry Measom in West Yorkshire, UK wrote:
"I also experienced quite dramatic front brake grab after complete brake overhaul. This only affected the left hand brake and only occurred on first or second application after being stood overnight. I assumed this was overnight rusting as I live in a damp climate but strange that it was only ever one brake. On disassembly the drum looked clean and rust free, but I noticed that the the ends of the friction material had varying degrees of chamfer. So I chamfered each end to approx 1/4 inch, and this seems to have cured the problem".