Abstract: La3.2Ba0.8Mn3O10, a representative of the rare n = 3 members of the Ruddlesden-Popper manganites A(n+1)Mn(n)O(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 MnO6 octahedra described within the symmetry-adapted mode approach (space group Cccm, a = 29.068(1) angstrom, b = 5.5504(5) angstrom, c = 5.5412(5) angstrom; PXD RF = 0.053, RP = 0.026; PED RF = 0.248). The perovskite block in La3.2Ba0.8Mn3O10 features an octahedral tilting distortion with out-of-phase rotations of the Mn06 octahedra according to the (Phi,Phi,0)(Phi,Phi,0) mode, observed for the first time in the n = 3 Ruddlesden-Popper structures. The Mn06 octahedra demonstrate a noticeable deformation with the elongation of two apical Mn-O bonds due to the Jahn-Teller effect in the Mn3+ 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 perovslcite and rock-salt blocks of the Ruddlesden-Popper structure are discussed. At low temperatures, La3.2Ba0.8Mn3O10 reveals a sizable remnant magnetization of about 1.3 mu(B)/Mn at 2K, and shows signatures of spin freezing below 150 K.