The J. C. Willems-Coppel-Shayman geometric characterization of solutions of the algebraic Riccati equation (ARE) is extended to asymmetric Riccati differential equations with time-varying coefficients. The coefficients do not need to satisfy any definiteness, periodicity, or system-theoretic condition. More precisely, given any two solutions X(1)(t) and X(2)(t) of such equation on a given interval [to, tl], we show how to construct a family of solutions of the same equation of the form X(t) = (I - pi(t))X(1)(t) + pi(t)X(2)(t), where pi is a suitable matrix-valued function. Even when specialized to the case of X(1) and X(2) equilibrium solutions of a symmetric equation with constant coefficients, our results condiserably extend the classical ones, as no further assumption is made on the pair X(1), X(2) and an the coefficient matrices.