As is well known, the law of parity conservation does not hold in weak interactions. This type of asymmetry created a number of theoretical problems which were solved, first of all, by a new understanding of the features of neutrinos and their role in weak interactions. These solutions were based, however, essentially on the handedness (chirality) of neutrinos which is closely related to their vanishing rest mass. The thesis of neutrinos with nonvanishing rest mass, newly considered in the literature, therefore requires a rediscussion of the early arguments concerning the relation between the neutrino theory and some weak interaction essentials. When one does this, as in the present paper, it is noted that neutrinos with rest mass lead to some difficulties, in particular to a violation of T invariance.