  We study the effective symmetry profiles of superconducting pair correlations
and the flow of charge supercurrent in ballistic Weyl semimetal systems with a
tilted dispersion relation. Utilizing a microscopic method in the ballistic
regime and starting from both opposite-pseudospin and equal-pseudospin
phonon-mediated opposite-spin electron-electron couplings, we calculate the
anomalous Green's function to study various superconducting pair correlations
that Weyl semimetal systems may develop. The momentum-space profile reveals
that by properly manipulating the parameters of Weyl semimetal systems,
including the tilting parameter, the effective symmetry class of even-parity
s-wave (odd-parity p-wave) superconducting correlations can be converted into a
d-wave (f-wave) symmetry class that consists of equal-pseudospin and
opposite-pseudospin channels. We also find that the supercurrent in a ballistic
Weyl Josephson junction can be made to vanish or switch directions, depending
on the tilt of the Weyl cones, in addition to the relevant parameters
characterizing the Weyl semimetal and junction. We show that inversion symmetry
breaking terms introduce transitions that result in the appearance of
self-biased current at zero difference between the macroscopic phases of the
superconducting segments, creating a phi0 Josephson state. Weyl semimetal
systems are shown to offer several experimentally tunable parameters to control
the induction of higher harmonics into the current phase relations.
