Purely antiferromagnetic oscillations (antimagnons) in tetragonal magnets with the crystal-chemical structure of the trirutiles (Fe2TeO6, etc.) are investigated theoretically. These oscillations were considered by Ukrainian physicists back in 1988, but not for tetragonal crystals and only in the case of an antiferromagnetic (AF) structure. In the present study, besides the AF phases the ferromagnetic phase in the same system of four magnetic sublattices is also considered. Ways of exciting the antimagnons, which are a fourth form of spin waves (in addition to the ferromagnons, quasiferromagnons, and quasiantiferromagnons), are also discussed. Independently of the means of excitation (electric or magnetic field, hypersound), antimagnons are defined as spin waves for which the total local ferromagnetic vector M(r) does not take part in the oscillations, but only the AF vector (or vectors) L(r). The eigenfrequencies and the corresponding susceptibilities are calculated. The presentation of this material is p...