  The defining conditions for the irreducible tensor operators associated with
the unitary irreducible corepresentions of compact quantum group algebras are
deduced first in both the right and left regular coaction formalisms. In each
case 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. Also included are discussions (within the regular coaction
formalisms for compact quantum group algebras) of inner-products, basis
functions, projection operators, Clebsch-Gordan coefficients, and two types of
tensor product of corepresentations. The formulation of quantum homogeneous
spaces for compact quantum group algebras is discussed, and the defining
conditions for the irreducible tensor operators associated with such quan!
  tum homogeneous spaces and with the unitary irreducible corepresentions of
the compact quantum group algebras are then deduced. There are two versions,
which correspond to restrictions of the right and left regular coactions. In
each case it is again shown that there are ordinary and twisted irreducible
tensor operators. Various consequences are deduced, including the corresponding
generalizations of the Wigner-Eckart theorem.
