The article addresses multivariate interpolation in the presence ofsymmetry. Interpolation is a prime tool in algebraic computationwhile symmetry is a qualitative feature that can be more relevantto a mathematical model than the numerical accuracy of the pa-rameters. The article shows how to exactly preserve symmetryin multivariate interpolation while exploiting it to alleviate thecomputational cost. We revisit minimal degree and least interpo-lation with symmetry adapted bases, rather than monomial bases.This allows to construct bases of invariant interpolation spaces inblocks, capturing the inherent redundancy in the computations.We show that the so constructed symmetry adapted interpolationbases alleviate the computational cost of any interpolation problemand automatically preserve any aquivariance of thir interpolation problem might have.