Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of (X, Y )-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).