Perovskites like LaAlO3, (or SrTiO3) undergo displacive structural phase transitions from a cubic crystal to a trigonal (or tertagonal) structure. For many years, the critical exponents in both these types of transitions have been fitted to those of the isotropic three-commponents Heisenberg model. Recent field theoretical accurate calculations showed that this is wrong: the isotropic fixed point of the renormalization group (RG) is unstable, and RG iterations flow either to a `cubic' fixed point or to a fluctuation-driven first-order transition. These distinct flows correspond to two distinct universality classes, identified by the symmetry of the ordered structures below the transitions. Here we show that perovskites which become trigonal or tetragonal belong to these two universality classes, respectively. The close vicinity of the isotropic and cubic fixed points explains the apparent wrong observations of a single universality class, but also implies the existence of slowly varying effective material-dependent exponents. For the tetragonal case, these effective exponents can have the `isotropic' values before crossing to the first-order transition. We propose dedicated experiments to test these predictions. We also expect a similar splitting of the universality classes in any situation in which two (or more) fixed points compete for stability.