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In this work, we systematically study two phases, called Andreev $\pi$-phaseand orbital-phase, and their influence on the Josephson effect. When the systemis time-reversal invariant and centrosymmetric, these two phases only appear inthe odd-parity pairings. The Andreev $\pi$-phase has nothing to do with thespecific form of the odd-parity pairings and means an intrinsic $\pi$-phasebetween the spin-triplet Cooper pairs entering and leaving CTSCs in the Andreevreflections. The orbital-phase corresponds to the phase difference between thespin-triplet Cooper pairs with opposite spin polarization and depends on thespecific form of the odd-parity gap functions. When the normal region of theJosephson junction contacts the same side of the CTSCs with some specificodd-parity parings, the competition between the two phases can lead to theJosephson $\pi$-junction. Note that this junction is different from that of theconventional Josephson junction (JJ) and is dubbed a U-shaped junctionaccording to its geometry. Meanwhile, in a conventional JJ, the interplay ofthese two phases causes their impact on the CPR to be completely canceled out.Therefore no matter what kind of pairing symmetries the CTSC has, it will leadto Josephson 0-junction in this case. We obtain our results based on the modelof the M$_{x}$Bi$_2$Se$_3$ family where M may be Cu, Sr, or Nb. Therefore, wepropose to detect the pairing symmetry of M$_{x}$Bi$_2$Se$_3$ through asuperconducting quantum interference device containing a U-shaped Josephsonjunction.

Intrinsic Andreev $\pi$-reflection and Josephson $\pi$-junction for centrosymmetric spin-triplet superconductors