The two dominant features in the distribution of orbital parameters for close-in exoplanets are the prevalence of circular orbits for very short periods, and the observation that planets on closer orbits tend to be heavier. The first feature is interpreted as a signature of tidal evolution, while the origin of the second, a 'mass-period relation' for hot Jupiters, is not understood. In this paper we reconsider the ensemble properties of transiting exoplanets with well-measured parameters, focusing on orbital eccentricity and the mass-period relation. We recalculate the constraints on eccentricity in a homogeneous way, using new radial velocity data, with particular attention to statistical biases. We find that planets on circular orbits gather in a well-defined region of the mass-period plane, close to the minimum period for any given mass. Exceptions to this pattern reported in the literature can be attributed to statistical biases. The ensemble data is compatible with classical tide theory with orbital circularization caused by tides raised on the planet, and suggest that tidal circularization and the stopping mechanisms for close-in planets are closely related to each other. The position mass-period relation is compatible with a relation between a planet's Hill radius and its present orbit.