Films of dipolar molecules formed by physical vapor deposition are, in general, spontaneously polarized, with implications ranging from electron transfer in molecular optoelectronic devices to the properties of astrochemical ices in the interstellar medium. Polarization arises from dipole orientation, which should intuitively decrease with increasing deposition temperature, T. However, it is experimentally found that minimum or maximum values in polarization vs T may be observed for cis-methyl formate, 1-propanol, and ammonia. A continuous analytic form of polarization vs T is developed, which has the property that it is not differentiable at all T. The minima and maxima in polarization vs T are marked by singularities in the differential of this analytic form. This exotic behavior is presently unique to films of dipolar species and has not been reported, for example, in the related magnetic phases of spin glasses.