  We demonstrate how the boundary-driven reconstruction of the superconducting
order parameter can be employed to manipulate the zero-energy Majorana bound
states (MBSs) occurring in a topological Josephson junction. We focus on an
interface of two p-wave superconductors, which are described by a spin-vector
order parameter $\bf{d}$. Apart from the sensitivity of $\bf{d}$ to external
Zeeman/exchange fields, here, we show that the orientation of $\bf{d}$
throughout the junction can be controlled by electrically gating the weak link.
The remarkable local character of this knob is a manifestation of the edge
reconstruction of the order parameter, which takes place whenever different
$\bf{d}$-vector configurations in each superconductor compete and are close in
energy. As a consequence, the spin-dependent superconducting-phase difference
across the junction is switchable from $0$ to $\pi$. Moreover, in the regime
where multiple edge MBSs occur for each superconductor, the Andreev-bound-state
(ABS) spectra can be twisted by the application of either a charge- or
spin-phase difference across the interface, and give rise to a rich diversity
of nonstandard ABS dispersions. Interestingly, some of these dispersions show
band crossings protected by fermion parity, despite their $2\pi$-periodic
character. These crossings additionally unlock the possibility of nontrivial
topology in synthetic spaces, when considering networks of such 1D junctions.
Lastly, the interface MBSs induce a distinct elecronic spin polarization near
the junction, which possesses a characteristic spatial pattern that allows the
detection of MBSs using spin-polarized scanning tunneling microscopy. These
findings unveil novel paths to mechanisms for ABS engineering and single-out
signatures relevant for the experimental detection and manipulation of MBSs.
