  It is shown that the azimuthal dependence of the distribution of hadrons in a
quark jet is a probe of the transverse spin of the quark initiating the jet.
This results in a new spin-dependent fragmentation function that acts at the
twist-2 level. One example of a process where it contributes is semi-inclusive
deeply inelastic scattering with a transversely polarized nucleon target but
with an {\em unpolarized} electron beam. This process is treated in detail.
Another process is the cross section for singly polarized hadron-hadron
scattering when two high $p_{\perp}$ hadrons are measured in the final state
and are close to back-to-back in azimuth. The new fragmentation function is
sensitive to the coupling of the fragmentation process to (spontaneous) chiral
symmetry breaking.
