  We study a Josephson junction involving an inversion-asymmetric Weyl
semimetal in presence of time-reversal symmetric (TRS) or time-reversal
symmetry broken tilt in the Weyl spectra. We reveal that both types of tilts in
the Weyl nodes lead to a Josephson $0$-$\pi$ transition and a zero bias
valley/chiral supercurrent. Strikingly, the TRS tilt gives rise to a pure
valley Josephson current (VJC) and TRS broken tilt induces a pure chirality
Josephson current (CJC) in this system. The VJC and CJC are the manifestation
of valley symmetry broken and $\mathbb{Z}_2$ symmetry broken by the respective
tilt. We obtain the reversal of a pure VJC and pure CJC even in the zero bias
condition controllable by the junction length. Our analysis of controllability
of valley and chirality dependent transport in an inversion asymmetric Weyl
semimetal junction could allow applications in valleytronics and
chiralitytronics, respectively. The tilt induced Josephson effect provides an
alternative route for supercurrent $0$-$\pi$ transition, different from the
conventional ferromagnetism Josephson junctions where the spin polarization is
essential. In the long junction and zero temperature limit, VJC and CJC are
associated with a quantum anomaly which is manifested through a discontinuous
jump in the current in absence of TRS and TRS breaking tilts, respectively.
