This type of pericyclic reaction involves the movement of a sigma bond, hence the name.  If these reactions are to occur in a concerted and cyclic fashion, then, as with other pericyclic reactions, we must consider the orbital symmetry involved.

Hydrogen Shifts

We can view the transition state of such a migration as being made up of the hydrogen atom radical, and the radical of the rest of the molecule.  For example:

Now we need to look at the relative phases of the terminal lobes of the HOMO of the cyclopentadienyl radical to see if migration is permitted.  We know that as there are 5 pi electrons in the radical, the HOMO must be Ψ₃.  Therefore, we can deduce that the end lobes look like this:

Now we can see that both terminal lobes, and the hydrogen radical lobe are all of the same phase.  This is critically important, as this means that all these interactions are bonding ones, and hence this type of migration (known as a 1,5  sigmatropic shift because of the starting and ending positions of the hrdrogen) can take place.  If, however, we had chosen a 1,4 sigmatropic shift, we would find that it is symmetry forbidden, and hence these shifts do not occur.  Indeed, thermal 1,5 shifts are quite common, and 1,4 shifts almost never occur.

Carbon Shifts

For example

Cope Rearrangement:

or

Claisen Rearrangement:

Generally, thermal carbon shifts proceed most readily via 6 membered transition states which can adopt a chair formation.  The same symmetry considerations apply as in hydrogen shifts.