  T-duality in M(atrix) theory has been argued to be realized as Morita
equivalence in Yang-Mills theory on a non-commutative torus (NCSYM). Even
though the two have the same structure group, they differ in their action since
Morita equivalence makes crucial use of an additional modulus on the NCSYM
side, the constant Abelian magnetic background. In this paper, we reanalyze and
clarify the correspondence between M(atrix) theory and NCSYM, and provide two
resolutions of this puzzle. In the first of them, the standard map is kept and
the extra modulus is ignored, but the anomalous transformation is offset by the
M(atrix) theory ``rest term''. In the second, the standard map is modified so
that the duality transformations agree, and a $SO(d)$ symmetry is found to
eliminate the spurious modulus. We argue that this is a true symmetry of
supersymmetric Born-Infeld theory on a non-commutative torus, which allows to
freely trade a constant magnetic background for non-commutativity of the
base-space. We also obtain a BPS mass formula for this theory, invariant under
T-duality, U-duality, and continuous $SO(d)$ symmetry.
