  We discuss the construction of $\kappa$-Poincar\'e invariant actions for
gauge theories on $\kappa$-Minkowski spaces. We consider various classes of
untwisted and (bi)twisted differential calculi. Starting from a natural class
of noncommutative differential calculi based on a particular type of twisted
derivations belonging to the algebra of deformed translations, combined with a
twisted extension of the notion of connection, we prove an algebraic relation
between the various twists and the classical dimension d of the
$\kappa$-Minkowski space(-time) ensuring the gauge invariance of the candidate
actions for gauge theories. We show that within a natural differential calculus
based on a distinguished set of twisted derivations, d=5 is the unique value
for the classical dimension at which the gauge action supports both the gauge
invariance and the $\kappa$-Poincar\'e invariance. Within standard (untwisted)
differential calculi, we show that the full gauge invariance cannot be
achieved, although an invariance under a group of transformations constrained
by the modular (Tomita) operator stemming from the $\kappa$-Poincar\'e
invariance still holds.
