How do permanent magnets manage to focus field on one side? The actuator of a hard drive head consists of two very strong neodymium magnets, with an electromagnetic coil between them. If you take that apart, the magnets attract each other very strongly. There's no doubt the field between them is very strong.
But if you flip them back to back, there is no repulsion force - there is pretty much nothing. While the magnets are very strong on one side, they seem completely inert on the other.
I understand how a U-shape of a shaped magnet allows it to focus field near the two exposed poles, but how does a flat rectangular plate manage to keep the field over one flat, wide surface and nearly none near the other?
[sorry about the heavy edit but it seems the question got totally muddled with irrelevant discussion about unipolar magnets and possibility or impossibility to remove magnetic field from one side. Only heavy rephrasing may help.]
Edit: some photos:
1: The magnets stuck together. They hold really hard, I'd be unable to just pull them apart but I can slide one against the other to separate them.

The magnets in "inert position" - they don't act on each other at all, just like two inert pieces of metal.

The magnets seem to have two poles located on the surface off-center. Here, a normal magnet placed against the hard disk magnet, centering itself on one side, then flipped - on another.
  
The metal shield seems to act like completely unmagnetized ferromagnetic. I can stick the "normal magnet" any way I like to it, and it doesn't act on another piece of ferromagnetic (a needle) at all.

When I apply a small magnet to it, it becomes magnetic like any normal "soft" ferromagnetic - attracts the needle weakly. It behaves as if the (very powerful) neodymium magnet glued to the other side wasn't there at all.

Unfortunately the neodymium magnets are glued very well and so fragile I was unable to separate any without snapping it, and then the "special properties" seem to be gone.
 A: I think that you are mistaking "one-sided" magnets and "unipolar" magnets. 

...strictly one-sided magnet (generating no magnetic field whatsoever on one side) is impossible...

No. There is no problem in having a magnet with a field on one side. The hdd head is the example (here is a picture). 
What you cannot have is a magnet with only one pole -- only S or only N. So, to claim that you have an "impossible" or "nearly impossible" magnet, you need not just to see if it "attracts metallic objects on one side" -- that works even with basic U-shaped magnets (some pictures here).  What you need to check is that you have only one magnetic pole all over the magnet.
A: A Halbach array? Basically the magnetic domains are oriented in a pattern that produces the appearance of a strong magnetic field on one side and a significantly reduced magnetic field on the other side.
http://en.wikipedia.org/wiki/Halbach_array
A: The metal plates the magnets are glued to are an Iron and Nickel alloy that has a very high magnetic permeability called a Mu-metal.
I don't understand all of the details of magnetism or how Mu-metal works but that should get you started.
A: The field appears to be unipolar but this is because both poles are on the same side of the magnet. This is the reason that the test magnets snap into a special configuration -- opposite poles attract, and the test magnet also has a "north" and "south" end.
I am only repeating what the other commenters have said, that is, that this is like a U-magnet or a (very) primitive Halbach array.
If you hold two U-magnets "butt-to-butt" they will not attract. This is because a U-magnet is just a bar magnet bent into a "U" shape, and the field lines for a bar magnet go from pole-to-pole. Therefore it makes sense why the field lines would be strongest near the poles, and not at the "butt."
When you squash the "butt" part flat, you get the weird magnet with a strong field on one side.
