These are stringent results on real homogeneous polynomials in several real variables: if the polynomials form a normalized biorthogonal system (with respect to a certain inner product, which is defined using partial derivatives of one of the components), then an addition theorem holds, and conversely: the addition property is sufficient for orthonormality, too. This property may be interpreted as Pythagorean identity, it generalizes earlier results of the first author [see Proc. Am. Math. Soc. 124, No. 7, 2001-2004 (1996, Zbl 0871.33010)]. Further, this is related to generating functions of polynomial solutions to partial differential equations [cf. the first author's work in SIAM J. Math. Anal. 22, No. 268-271 (1991; Zbl 0713.33004)]. Applications are the study of convergence radii of series of those polynomials and a Funk-Hecke theorem for homogeneous polynomials.