Why do two parts of a broken magnet repel each other?

When you break a magnet, it often seems that the polarities have flipped.

This doesn't actually happen.

No polarity reversal occurs when you think the poles have flipped

- you are dealing with a magnet that has an axial field 

(pointing out through the flat face.) When you break 
it, each half has similar field, pointing in the same direction, which 
is unstable. One piece will want to flip so that the fields line up 
anti-parallel (lower energy situation).  

Whether the broken magnet attracts or repels each other depends

on how the poles were previously present.

If the original magnet looked like
  +-----------------------------+
N |                             | S
  +-----------------------------+ 

After it's broken, it becomes
  +-----------+     +---------------+
N |           |S   N|               | S
  +-----------+     +---------------+ 

The two broken parts will attract each other.

However, if the original magnet looked like:

(poles on the flat faces of the magnet)

                N
  +--------------------------------+
  |                                |
  +--------------------------------+
                S

After it is broken, it becomes


         N                    N
  +---------------+   +-----------------+
  |               |   |                 |
  +---------------+   +-----------------+
         S                    S

Therefore the two parts will repel each other and will try to invert.

These are only two simple cases. In reality, the poles
of a magnet can be more complicated. This just illustrates how the poles
affect behavior of magnets.