In this paper we study the competition between helical and localizing modes in the torsional buckling of stretched and twisted elastic rods. Within the Love-Kirchhoff formulation, we make comparative studies of the helical deformation of Love (1927) and the homoclinic localizing solution of Coyne (1990). Plots of the loads against their corresponding deflections allow the energetically preferred mode to be identified: it is found that preference switches from the helix to the localized mode early in the post-buckling range. These plots also allow us to predict the jumps that are observed under a variety of dead and rigid loading processes: these dynamic jumps take the rod from the spatially localized form to the familiar writhing state. Preliminary experiments confirm this preference for the localizing mode. They also reveal a second type of helical deformation, at a shorter wavelength, that is not predicted by the above long-rod analyses. A programme of further experimental and theoretical studies is suggested, and in a companion paper we lay the mathematical foundations for a numerical investigation of the complex and spatially chaotic deformations of a wider class of elastic rods.