Symmetries are essential for a consistent formulation of many quantum systems. In this paper we discuss a fundamental symmetry, which is present for any Lagrangian term that involves ẋ2. As a basic model that incorporates the fundamental symmetries of quantum gravity and string theory, we consider the Lagrangian action of the relativistic point particle. A path integral quantization for this seemingly simple system has long presented notorious problems. Here we show that those problems are overcome by taking into account the additional symmetry, leading directly to the exact Klein-Gordon propagator.