  The interaction mediated by irreducible pion and gluon exchange between
constituent quarks is calculated and shown to have a strong tensor component,
which tends to cancel the pion exchange tensor interaction between quarks. Its
spin-spin component is somewhat weaker than the pion exchange spin-spin
interaction, while its central and spin-orbit components are small in
comparison to the corresponding single gluon exchange interactions. The
combination of the $\pi-$gluon exchange interaction with the single pion
exchange interaction and a weak gluon exchange interaction between constituent
quarks has the qualitative features required for understanding the hyperfine
splittings of the spectra of the nucleon and the $\Delta$ resonances.
