Modules of principal parts on the projective line

We study the splitting-type of the bi-modules of principal parts (Grothendiecks analogue of jet-bundles in algebraic geometry) as left and right O-module on the projective line in positive characteristic, and obtain explicit examples where the splitting-type as left O-module differs from the splitting-type as right O-module. Hence the principal parts are the first examples of a sheaf of abelian groups equipped with two non-isomorphic structures as locally free O-module. We also give explicit examples showing that in some cases the splitting-type changes with the characteristic of the base-field.