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The defining conditions for the irreducible tensor operators associated withthe unitary irreducible corepresentions of compact quantum group algebras arededuced first in both the right and left regular coaction formalisms. In eachcase it is shown that there are {\em{two}} types of irreducible tensoroperator, which may be called `ordinary' and `twisted'. The consistency of thedefinitions is demonstrated, and various consequences are deduced, includinggeneralizations of the Wigner-Eckart theorem for both the ordinary and twistedoperators. Also included are discussions (within the regular coactionformalisms for compact quantum group algebras) of inner-products, basisfunctions, projection operators, Clebsch-Gordan coefficients, and two types oftensor product of corepresentations. The formulation of quantum homogeneousspaces for compact quantum group algebras is discussed, and the definingconditions for the irreducible tensor operators associated with such quan!  tum homogeneous spaces and with the unitary irreducible corepresentions ofthe compact quantum group algebras are then deduced. There are two versions,which correspond to restrictions of the right and left regular coactions. Ineach case it is again shown that there are ordinary and twisted irreducibletensor operators. Various consequences are deduced, including the correspondinggeneralizations of the Wigner-Eckart theorem.Comment: To be published in J.Math.Phys., 70 pages, RevTe

Irreducible tensor operators in the regular coaction formalisms of compact quantum group algebras