I figured I ought to update this thread one last time.

Thanks to everyone that replied with suggestions and information. I rearranged the brake shoes and springs on both wheels and adjusted the brakes. We recently had our MOASF Banzai Beach Bash run last Saturday and I am happy to report that I only noticed a bit of "grab and release" of the rear brakes for a short time on the way up the run start point. After that I didn't feel any of the symptoms! So, it's cured.

Thanks again for the multiple second set of eyes and helpful suggestions. You folks are the best!

Amazingly enough, they are front brake shoes. But only for single leading shoe (early) Minis. The other front brake shoes listed are 1.5 inch wide and for use on twin leading shoe (late non-disc) Minis. The single leading shoes are 1 1/4 inch wide and are used on all rear Mini brakes from what I can find. Also on Midgets I think!

Those look like front shoes.

Another thing I notice is that it looks like the shoes were installed without doing a little chamfer on the front and rear of the pads before installation. That helps break them in when first put in use.

Took it out for a drive and the problem is still there, but not as pronounced as before I fixed the brake arrangement.

Just took a look at the photos. As far as the shoes were concerned, the left rear was set up correctly. The right rear was not.

I have rearranged the shoes and will take it out for a drive. Not expecting much of a change though. When I adjust the brakes they rub the drums and then let go. This only happens once each rotation of the drum. I would expect warped drums to do that twice. I measured across the ID of each drum with digital calipers in many positions and get the same number at all points. It seems hub related to me, but the grab and release part is baffling.

I have reached the point where I only use Mintex rear brake shoes. They come much closer to the radius of the drums. Steve (CTR)

The things I noticed may not be the root cause of your car's problem.  However, I would address them so you can exclude them. 

Keep us posted on your progress.

Thanks Doug. Wow, how could I have put those back on the car in a more messed up way. The right side forward shoe is...

Ah heck, they're just all wrong. I gotta stop drinkin' Red Stripe and working on cars! (wink, wink, nudge, nudge). I'll be changing them around tomorrow after the cars and coffee and before our Bonzai Beach Bash run. Did I mention that MOASF's annual kick off the year run is the Saturday after this?

Thanks again all. Sometimes I just have to stand back and have someone else take a look at the mess I've made.

Craig in Campbell

Are you using the right shoes? I know front drum brake shoes are wider than rears, 1 1/2 " versus 1 1/4", I believe.

Mark commented about the shoe placement.  It can make a difference but usually only to the "feel" of the brakes.  I am posting a picture link below showing a right, rear backplate assembly.  Take a look at it and compare it to the brakes on your car.  (Cut and paste the image address into a new window/tab if necessary so you can look at a larger version of the brake assembly picture).

Notice the following:
1) Look at the length of the friction material on the metal shoe.  You will see the friction material is relieved on one end of each shoe.  Where that relief is relative to the direction of wheel rotation can make a difference in pedal feel and the ability of the shoes to conform to the inside of the drum.  To me it looks like your shoes are not on in the correct position.  Note the picture shows the right rear.  The left rear is the mirror image.
2) Look at the springs.  It looks like you have placed the springs on the wheel flange side of the shoes while the picture shows the springs on the "inside" behind the shoes and closer to the centerline of the car.  If the springs are on the wrong side the shoes tend to "lean out" as is shown in your pictures.

I would start by moving the shoes so the friction material is in the right place and put the springs on correctly.

" www.minimania.com/images_temp/80060000121a1058kit.jpg " (RIGHT REAR brake backplate assembly).

Also see the Moke LEFT rear brake assembly in the picture link below.  This picture shows in more detail where the friction material is relieved.  Remember in the picture link below that this is the LEFT side and the wheel rotates counterclockwise relative to the brake assembly.  This is the mirror image of the right rear plate shown above.

//www.reizendemoke.be/techniek/manual/brakes/reardrum.gif

So, hopefully, there are a few photos in this post. They are the Right Forward Shoe, Right Aft Shoe, Right Brake Setup, Left Forward Shoe, Left Aft Shoe and Left Brake Setup. It looks to me like the shoes are not engaging the drums evenly. Feel free to throw rocks or kick tires.

Take a photo of your shoe placement, too. There is a right and wrong way, although many have said it (usually) doesn’t matter. Maybe it does.

I usually don't set the e-brake after driving hard for just that reason. This problem was there the first time out with the drums when they were new and after machining. I will pull the drums off today and see if there is any wear on the backing plates and check on the wear patterns inside the drums.

Parking the car with hot brakes and setting the handbrake can warp drums out of round.
I'd also check the drums and back-plates for rubbing of the shoe edges/sides.

Occurs not when cold but only when warm may be the clue. Are your shoe pads expanding? I had a set that was time-of-year-moisture sensitive and I had to back off the adjusters 3 flats. Pad materials are strange stuff these days. Maybe yours are heat sensitive, expanding, dragging, heating up and warping the drum. It cost nothing but a little time to 
adjust both rears looser. Do the experiment and see what happens.

I run a late master cyl./booster with 7.5 Cooper S front brakes and drums ( super fins ) at the rear. I started with 3/4 wheel cyls. and even with twin tanks the rears would lock up first made worse by a wet or sandy surface. I either changed to 1/2 or 5/8 rear cyls. After 4 years of daily driving I'm down to rivets on my rear shoes. They must be working. I have also found the better quality shoes are the only choice. The cheap ones do not have the correct radius. Steve (CTR)

and you have the correct wheel cylinders for your application?