  Defining conditions for irreducible tensor operators associated with the
unitary irreducible corepresentations of compact quantum group algebras are
deduced within the framework of the abstract carrier space formalism. It is
shown that there are {\em{two}} types of irreducible tensor operator, which may
be called `ordinary' and `twisted'. The consistency of the definitions is
demonstrated, and various consequences are deduced, including generalizations
of the Wigner-Eckart theorem for both the ordinary and twisted operators.
Examples of irreducible tensor operators for the standard deformation of the
function algebra of the compact Lie group $SU(2)$ are described to demonstrate
the applicability of the new definitions.
