Introduction. The role of integral operators with homogeneous kernels in mathematics and applications is well-known. Now we have a number of useful results concerning the operators, whose kernels are homogeneous of the degree (−n) and invariant with respect to the rotation group SO(n). Namely, the criteria of the invertibility and the Noether property are obtained; the Banach algebras generated by these operators are described; the conditions for the applicability of the projection method (see, for instance, [1]–[5] and references therein) are established. In this paper, we consider multidimensional integral operators, whose kernels have a mixed homogeneity: in a part of variables the kernel is homogeneous of the degree (−n), and in the rest variables the kernel is coordinate-wise homogeneous. For these operators below we prove the boundedness theorem and establish the invertibility criterion. Below we use the following notation: Rn is the n-dimensional Euclidean space; x = (x1, . . . , xn) ∈ Rn; |x| = √ x1 + · · · + xn; x′ = x/|x|; x · y = x1y1 + · · ·+ xnyn; x ◦ y = (x1y1, . . . , xnyn); e1 = (1, 0, . . . , 0); Sn−1 = {x ∈ Rn : |x| = 1}; R+ = {x ∈ Rn : xj > 0 ∀j = 1, 2, . . . , n}; Z+ is the set of all integer nonnegative numbers; Yνμ(σ) are spherical harmonics of the order ν (∈ Z+);