La 3.2Ba 0.8Mn 3O 10, a representative of the rare n = 3 members of the Ruddlesden-Popper manganites A n+1Mn nO 3n+1, was synthesized in an evacuated sealed silica tube. Its crystal structure was refined from a combination of powder X-ray diffraction (PXD) and precession electron diffraction (PED) data, with the rotations of the MnO 6 octahedra described within the symmetry-adapted mode approach (space group Cccm, a = 29.068(1) Å, b = 5.5504(5) Å, c = 5.5412(5) Å; PXD R F = 0.053, R P = 0.026; PED R F = 0.248). The perovskite block in La 3.2Ba 0.8Mn 3O 10 features an octahedral tilting distortion with out-of-phase rotations of the MnO 6 octahedra according to the (φ,φ,0)(φ,φ,0) mode, observed for the first time in the n = 3 Ruddlesden-Popper structures. The MnO 6 octahedra demonstrate a noticeable deformation with the elongation of two apical Mn-O bonds due to the Jahn-Teller effect in the Mn 3+ cations. The relationships between the octahedral tilting distortion, the ionic radii of the cations at the A- and B-positions, and the mismatch between the perovskite and rock-salt blocks of the Ruddlesden-Popper structure are discussed. At low temperatures, La 3.2Ba 0.8Mn 3O 10 reveals a sizable remnant magnetization of about 1.3 μ B/Mn at 2 K, and shows signatures of spin freezing below 150 K.