Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(x i), i = 1(1)2 n+1, gives a trigonometric sum of the n th order, S 2 n+1 ( x = a 0 + ∑ j n = 1 (a jcos jx + b jsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S 2 r+1 ( x) is one that satisfies ∫ a bS 2r+1(x)S 2r′+1(x)dx=0, r′<r and two other arbitrarily imposable conditions needed to make S 2 r1 ( x) unique. Two proofs are given of a fundamental factor theorem for any S 2 n+1 ( x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [ n/2] + 1, which are exact for any S 4 r−1 ( x). We have ∫ a bS 4r−1(x)dx=∑ j=1 2rA jS 4r−1(x j) where the nodes x j, j = 1(1)2 r, are the zeros of the orthogonal S 2 r+1 ( x). It is proven that A j > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S 2 r+1 ( x), x j and A j , j = 1(1)2 r, were calculated for [ a, b] = [0, π/4], 2 r = 2 and 4, to single-precision accuracy.