Spline interpolation requires the definition of boundary condition because the smoothness requirements do not yield enough conditions for a unique solution.

Question:

which kind of boundary conditions guarantee that the interpolating spline reproduces a sampled polynomial if its degree isn't higher than that of the interpolating spline, i.e. yield the algebraically simplest interpolating spline functions?

Natural Cubic Splines are a counter example, because a cubic polynomial has exactly one inflection point, whereas a cubic natural spline has at least two.