  Characterizing the duals of linear codes with rich algebraic structures
received great interest in recent decades. The beginning was by representing
cyclic codes over finite fields as ideals in the polynomial ring. Subsequently,
studying the duals of constacyclic, quasi-cyclic, quasi-twisted, generalized
quasi-cyclic, and multi-twisted codes appeared extensively in literature. We
consider the class of multi-twisted (MT) codes because it extends to all of
these codes. We describe a MT code $\mathcal{C}$ as a module over a principal
ideal domain. Hence, $\mathcal{C}$ has a generator polynomial matrix (GPM) that
satisfies an identical equation. The reduced GPM of $\mathcal{C}$ is the
Hermite normal form of its GPM. We show that the Euclidean dual
$\mathcal{C}^\perp$ of $\mathcal{C}$ is MT as well. We prove a formula for a
GPM of $\mathcal{C}^\perp$ using the identical equation of the reduced GPM of
$\mathcal{C}$. Then we aim to replace the Euclidean dual with the Galois dual.
The Galois inner product is an asymmetric form, so we distinguish between the
right and left Galois duals. We show that the right and left Galois duals of a
MT code are MT as well but with possibly different shift constants. Our study
is the first to contain the right and left Galois duals of a linear code
simultaneously. This gives two advantages: establishing their interconnected
identities and introducing the two-sided Galois dual that has not previously
appeared in the literature. We use a condition for the two-sided Galois dual of
a MT code to be MT, hence its GPM is characterized. Two special cases are also
studied, one when the right and left Galois duals trivially intersect and the
other when they coincide. The latter case is considered for any linear code,
where a necessary and sufficient condition is established for the equality of
the right and left Galois duals.
