<p>We establish a new family of the critical higher order Sobolev interpolation inequalities for radial functions as well as for non-radial functions. These Sobolev interpolation inequalities are sharp in the sense that they use the optimal quadratic forms of the sharp Hardy-Rellich inequalities and cover the Sobolev critical exponents. Our results extend those studied by Dietze and Nam in (Calc Var Partial Differ Equ 63:184, 2024) for the first order derivative case to higher order setting. The well-known Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of Dietze and Nam (Calc Var Partial Differ Equ 63:184, 2024). To overcome the absence of the Pólya-Szegö principle and the nonlinear ground state representation in the higher order case, our proofs rely on the Fourier analysis and a higher order version of the Talenti comparison principle. We also study a new version of the critical Hardy-Sobolev interpolation inequality involving the critical quadratic form of the Hardy inequality and Lorentz norms. Our critical Hardy-Sobolev interpolation inequality complements the result of Dietze and Nam in (Calc Var Partial Differ Equ 63:184, 2024).</p>