  A color-singlet neutral vector boson is considered as a possible common
origin of the top forward-backward asymmetry and the CDF dijet resonance. We
identify chiral and flavor structures of the couplings of this new vector boson
to the standard model quarks for which one could accommodate both data. We also
demonstrate that non-vanishing observables involving longitudinal top
polarizations can provide useful criteria for the possible existence of parity
violating new physics in the q q^bar -> t t^bar process, and discrimination
between the flavor-conserving and flavor-violating cases.
