By use of a local stability criterion recently introduced, we predict the existence of a periodic saddle-splay Freedericksz (PSSF) transition that adds to the existing class of classical Freedericksz transitions driven in a nematic cell by an external field. Occurrence of the PSSF transition requires a saddle-splay elastic constant with a large enough magnitude and different anchoring strengths at the plates confining the nematic cell. Otherwise, either the PSSF transition does not occur at all, or it requires a field higher than that associated with the classical aperiodic splay Freedericksz (ASF) transition, in which case it is not observable. Here, we determine the threshold field for which the PSSF precedes the ASF transition, as well as the structure of the destabilizing mode.