Here, by combining a symmetry-based analysis with numerical computations, we predict a different kind of magnetic ordering - antichiral ferromagnetism. This term aims to reflect that spontaneous modulation of the magnetization direction m(r) appears in a way that both types of chirality (handedness) exist simultaneously, and alternate in space. Without loss of generality, we focus our investigation on crystals with full tetrahedral symmetry where chiral interaction terms - Lifshitz invariants - are forbidden by symmetry. However, we demonstrate that the leading chirality-related term leads to nontrivial smooth magnetic textures exhibiting antichirality. In addition to the unconventional ground state, the revealed ordering gives rise to rich phenomena such as unique magnetic domains and skyrmions.