A [1,j]-sigmatropic rearrangement is also a two component pericyclic reaction: one component is the π-system, the other component is the migrating group.  The simplest case is a [1,j]-hydride shift across a π-system where j is odd.  In this case, as the hydrogen has only a spherically symmetric s orbital, the reaction must be suprafacial with respect to the hydrogen.  The total number of electrons involved is (j + 1) as there are (j − 1)/2 π-bond plus the σ bond involved in the reaction.  If j = 4n − 1 then it must be antarafacial, and if j = 4n + 1, then it must be suprafacial.  This agrees with experiment that [1,3]-hydride shifts are generally not observed as the symmetry-allowed antarafacial process is not feasible, but [1,5]-hydride shifts are quite facile.
For a [1,j]-alkyl shift, where the reaction can be antarafacial (i.e. invert stereochemistry) with respect to the carbon center, the same rules apply.  If j = 4n − 1 then the reaction is symmetry-allowed if it is either antarafacial with respect to the π-system, or inverts stereochemistry at the carbon.  If j = 4n + 1 then the reaction is symmetry-allowed if it is suprafacial with respect to the π-system and retains stereochemistry at the carbon center.
On the right is one of the first examples of a [1,3]-sigmatropic shift to be discovered, reported by Berson in 1967.  In order to allow for inversion of configuration, as the σ bond breaks, the C(H)(D) moiety twists around at the transition state, with the hybridization of the carbon approximating sp, so that the remaining unhybridized p orbital maintains overlap with both carbons 1 and 3.