We re-examine the jet probes of the nucleon spin and flavor structures. We find for the first time
that the time-reversal odd (T-odd) component of a jet, conventionally thought to vanish, can survive
due to the nonperturbative fragmentation and hadronization effects. This additional contribution
of a jet will lead to novel jet phenomena relevant for unlocking the access to several spin structures
of the nucleon, which were thought to be impossible by using jets. As examples, we show how the
T-odd constituent can couple to the proton transversity at the Electron Ion Collider (EIC) and
can give rise to the anisotropy in the jet production in e+ e− annihilations. We expect the T-odd
contribution of the jet to have broad applications in high energy nuclear physics.