All diagonal proper Bianchi I space–times are determined which admit certain important symmetries. It is shown that for Homothetic motions, Conformal motions and Kinematic Self-Similarities the resulting space–times are defined explicitly in terms of a set of parameters, whereas Affine Collineations, Ricci Collineations, and Curvature Collineations, if they are admitted, they determine the metric modulo certain algebraic conditions. In all cases the symmetry vectors are explicitly computed. The physical and the geometrical consequences of the results are discussed and a new anisotropic fluid, physically valid solution which admits a proper conformal Killing vector, is given.