Multivariate Lagrange and Hermite interpolation are examples of ideal interpolation. More generally an ideal interpolation problem is defined by a set of linear forms, on the polynomial ring, whose kernels intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm.